On Gauge Invariance and Minimal Coupling

Aneesh V. Manohar; Elizabeth E. Jenkins; Michael Trott

REVIEW

On Gauge Invariance and Minimal Coupling

Elizabeth E. Jenkins, Aneesh V. Manohar, Michael Trott

High Energy Physics - Phenomenology · Submitted 30 Apr 2013 · 51 references

Pith's verdict: A correct and useful demolition of minimal-coupling operator classification, with one soft spot in the pseudo-Goldstone Higgs section where a naturalness relation is used as if it were a derivation.

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CitationAneesh V. Manohar, Elizabeth E. Jenkins, Michael Trott. "On Gauge Invariance and Minimal Coupling." arXiv:1305.0017 (2013). https://pith.science/paper/H3V7IVTF

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Editor's note

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The thing to know: this paper is right on the main point, and the main point matters. Minimal coupling is not a symmetry, so it cannot order operator coefficients in an EFT, and the authors show this cleanly rather than just asserting it. The Sec. 4 construction is the real contribution — take a vanishing ungauged operator like -φ†[∂,∂][∂,∂]φ, promote derivatives to covariant ones, and you land on e²φ†φFF, which the naive minimal-coupling prescription would have called loop-suppressed. Same ungauged Lagrangian, two different gauged answers. That is a genuine logical point, not a rhetorical one, and I have not seen it laid out this explicitly before. The non-Abelian version (O4 reachable, O5 not, because Tr[D,D][D,D] is meaningless without a field to act on) sharpens it further. Sec. 5 then does the empirical work: the neutron has q=0 and a magnetic moment of -1.91, hydrogen polarizability is 9a₀³/2 with no loops anywhere in sight, and L9, L10 in χPT come out slightly larger than their neighbors. Those are decisive against any claim that non-minimally-coupled operators are generically suppressed by g²/16π². The paper is also honest about lineage — it says plainly that Weinberg made the algebraic-versus-dynamical-symmetry distinction in 1970 and that the original PGB Higgs papers did not lean on minimal coupling. That honesty is worth noting; it is not a paper overselling its novelty. Where it is soft: Sec. 6.2. Eq. (6.18), m_H² ~ (g²Λ_S²/16π²)c₆, comes from closing the gauge loop in Fig. 1(b), which is a naturalness expectation, not an identity — in a general EFT the H†HFF coefficient and the Higgs mass term are independent. The authors do condition on the model's own assumption that the gauge loop generates the mass, so the argument is internally consistent, but the follow-on claim that c₆ is unsuppressed for Λ_S in the 1–10 TeV range reads as general when it is not, and the Little Higgs c₆ ~ g² statement is asserted rather than derived. Second, the critique of Refs. [13] and [14] would land harder with direct quotations; the paper reconstructs their intent rather than pinning it down. Neither of these touches the central claim. Who should read it: anyone building Higgs dimension-six analyses or composite-Higgs models who has absorbed the tree/loop folklore, and referees who keep seeing it invoked. Worth a serious referee report — the Sec. 6.2 overreach is exactly the kind of thing a careful report should force them to qualify.

Referee report

Significance

The paper addresses a genuine methodological question that had real consequences for the Higgs EFT literature of 2012–2013, and it does so with an unusually broad and checkable evidence base. The strongest asset is the set of empirical counterexamples in Sec. 5: the neutron magnetic moment (q = 0 yet μn = −1.91 nuclear magnetons, with the large-Nc relation μn/μp = −2/3 + O(1/Nc²) of Eq. (5.10)), the hydrogen electric polarizability αE = 9a0³/2 of Eq. (5.11), and the measured χPT coefficients L9, L10, which are order unity and in fact slightly larger than their neighbors. These are external, independently measured facts, not model assumptions, and they make the central claim falsifiable in the ordinary sense: any reader can check that neutral composite objects couple to field strengths without loop suppression. The Sec. 4 O2/O3 construction is a clean and reusable logical point, and the appendix toy model (Eqs. (A.1)–(A.2)) gives a constructive demonstration that operators conventionally labeled 'loop-level' can arise at tree level with order-one coefficients. The paper is also careful to note that the original pseudo-Goldstone Higgs papers [42–44] did not rely on minimal-coupling 2

Major objections (4)

[§6.2, Eqs. (6.17)–(6.19)] The quantitative claim that c6 is not loop suppressed rests on Eq. (6.18), m_H² ∼ (g²Λ_S²/16π²)c6, which is obtained by 'closing the gauge loop' in Fig. 1(b). In a general EFT the coefficient of H†H FμνFμν and the Higgs mass parameter are independent; Eq. (6.18) is a naturalness expectation that holds only if the gauge loop of Fig. 1(a) is the dominant source of m_H. The text does condition on this ('By assumption in some PGH models...'), but the subsequent conclusion, c6 ∼ (2.5 TeV/Λ_S)² and hence 'c6 is not suppressed by a loop factor for Λ_S in the 1−10 TeV range', is phrased as a general statement. A concrete test: in a model where m_H is generated dominantly by top loops, or by collective symmetry breaking with different dynamics, Eq. (6.18) does not fix c6 and the numerical bound does not follow. The authors should state the assumption explicitly at the point of Eq. (6.19) and separate the model-dependent numerical estimate from the model-independent statement that no symmetry forbids an unsuppressed c6.
[§6.2, Little Higgs sentence after Eq. (6.19)] The statement 'In Little Higgs theories, c6 would be order g², and m_H² order g⁴' is asserted without derivation, and it is the one place where the paper makes a model-specific quantitative claim about a named class of theories. Since this sentence is used to support the generality of the conclusion, it should either be derived (even at the level of a power-counting argument identifying which collective-symmetry-breaking operator yields c6 ∼ g²) or be softened to a statement about what Little Higgs power counting permits rather than what it predicts.
[§4 versus Refs. [13,14]] The ambiguity argument of Sec. 4 refutes minimal coupling read as an operator-level map from an ungauged Lagrangian to a gauged one. A reader sympathetic to Refs. [13,14] can reasonably reply that the intended statement is weaker: that the UV completion is a gauge theory whose own renormalizable Lagrangian contains only minimal couplings, with no non-minimal operators at the renormalizable level. Under that reading, Sec. 4 does not directly land, and the critique must be carried by the separate Sec. 7.3 argument that a UV theory cannot be tuned to enforce minimal coupling in the IR. The paper would be substantially stronger if it stated explicitly which reading of Refs. [13,14] it takes itself to be refuting, and noted that Sec. 7.3 is the argument that survives under the weaker reading. As written, the two strands are presented as parallel rather than as covering different readings of the target claim.
[§6.1.1, Eq. (6.10)] The claim that operators such as m_{u,d}(π⁰)²FμνFμν and m_s K⁰K̄⁰FμνFμν are generated at order p⁶ 'without any loop suppression', supported by the sentence 'We have explicitly verified that this is the case', is load-bearing: it is the QCD precedent on which the pseudo-Goldstone Higgs analogy of Sec. 6.2 rests. The verification is not displayed. The authors should either exhibit the relevant p⁶ operators and their coefficients (citing Refs. [39,40] where the p⁶ Lagrangian is given) or add a short appendix showing the matching, so that a reader can check the claim without reconstructing the p⁶ basis.

Minor objections (8)

[§7.2] The statement 'We have verified by explicit calculation that the cross section γ + W → γ + W in the EFT does not violate unitarity below Λ' is asserted without display; a sentence indicating the partial waves computed and the scaling with Λ would let the reader check it.
[§5.1] The observation that the gauge kinetic term GaμνGaμν and the θ-term 'violate minimal coupling' is correct as a definitional point but reads as a rhetorical flourish; one sentence acknowledging that these terms are not obtained from an ungauged theory at all, rather than being counterexamples to a prescription, would sharpen the paragraph.
[§6.2, Eq. (6.16)] In the c2 term, g⁴ Tr(U Ta U† Tb U Tc U† Tc), the index c is repeated inside a single trace and the color/flavor structure is not transparent; please check the index contractions and, if the expression is schematic, say so.
[Appendix A, Eq. (A.1)] The real scalar triplet is denoted Σa, while Σ(x) denotes the Goldstone field in the coset G/H in Sec. 6.2; a different symbol (e.g. χa or ηa) would avoid a notational clash between the appendix and the main text.
[§5.5, Eqs. (5.15)–(5.19)] The distinction between c and c̃ is introduced without a one-line reminder of why the relativistic coefficient c̃ differs from the physical moment by q; a short parenthetical at Eq. (5.15) would help the reader follow the matching argument.
[Footnote 7] Typo: 'The total π⁺ mass also gets a contribute from the u and d quark masses' should read 'contribution'.
[Table of contents, §7.2 heading] The heading renders as 'g⁄= 2' in the compiled text; please check the typesetting of the inequality in the section title.
[§5.3] The proton magnetic moment is quoted as c = 2.793 without units; adding 'in nuclear magnetons' at first use would make the comparison with the neutron value explicit.

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The paper

The paper claims

The paper argues that minimal coupling is not a well-defined principle for classifying higher-dimensional operators into tree-level and loop-level classes.

The reading

The paper sets out to show that minimal coupling—the rule that replaces ordinary derivatives by covariant derivatives to turn a global theory into a gauge theory—cannot serve as a general organizing principle for effective field theories. The authors argue that the rule is ambiguous at the operator level: algebraically equivalent ways of writing the same gauge-free Lagrangian can lead to different gauge-invariant operators once derivatives are promoted, so there is no unique split between 'minimally coupled' operators and operators that supposedly require a loop suppression. They illustrate the failure with quantum-mechanical magnetic moments, neutral composite states, chiral perturbation theory, and pseudo-Goldstone Higgs models. If they are right, many recent arguments that certain Higgs operators are suppressed by $g^2/(16π^2)$ are not generic predictions of an EFT, and the coefficients of field-strength Higgs operators should be treated as independent Wilson coefficients.

What carries the argument

The load-bearing device is the commutator of covariant derivatives, $[D_μ,D_ν]=igF_{μν}$ (with the appropriate representation matrices), which vanishes when gauge fields are turned off but produces field-strength operators when they are turned on. The authors use this to construct two inequivalent minimal-coupling prescriptions—ordinary minimal coupling and 'next-to-minimal coupling'—that assign different tree-versus-loop labels to the same gauge-invariant operator. They then deploy examples where non-minimal couplings appear with order-unity coefficients: the electron and proton magnetic moments, the neutron's non-zero magnetic moment despite zero charge, Hydrogen polarizability, quarkonium couplings to chromoelectric and chromomagnetic fields, the $L_9$ and $L_{10}$ terms in chiral perturbation theory, and the operators $O_{WΣ}$ and $O_{BΣ}$ in pseudo-Goldstone Higgs models.

Extended reading notes

Core claim

The central claim is that minimal coupling is ill-defined as an operator-level classification scheme. The authors show that the same ungauged operator can be written in forms that vanish identically, such as $-φ^†[∂_μ,∂_ν][∂^μ,∂^ν]φ$, but that become non-zero and different after the replacement $∂→D$, because $[D_μ,D_ν]∝F_{μν}$. This lets one 'derive' $e^2φ^†φF_{μν}F^{μν}$ from a vanishing starting point, while the analogous neutral-field operator $e^2Φ^†ΦF_{μν}F^{μν}$ cannot be obtained at all; the authors call the first route next-to-minimal coupling. They extend the point to non-Abelian fields, quantum mechanics, bound states, and chiral Lagrangians, and conclude that non-minimally-coupled operators need not carry a loop factor.

Load-bearing premise

The argument assumes that the 'minimal coupling' classification being criticized is meant as an operator-level rule for mapping a gauge-free Lagrangian to a gauged one; if the claim is only that a renormalizable UV completion has no non-minimal couplings at its own leading order, the ambiguity examples do not by themselves settle the IR question.

Editorial extensions

If this is right

Dimension-six Higgs operators containing $W_{μν}$, $B_{μν}$, or $G_{μν}$ should be included with independent coefficients in EFT fits; no automatic $g^2/(16π^2)$ suppression follows from their non-minimal coupling.
The renormalization-group mixing of these operators into $h→γγ$ and $h→Zγ$ can produce observable deviations even when the underlying UV theory is strongly interacting and no loop suppression is assumed.
Pseudo-Goldstone Higgs models generically contain operators such as $O_{WΣ}$ and $O_{BΣ}$, and the coefficient of $H^†HF_{μν}F^{μν}$ is tied to the Higgs mass by a closing-the-loop relation, not forced to be loop-small.
Operator bases related by equations of motion are physically equivalent, so a 'special basis' cannot by itself create a physical tree/loop classification.
Neutral bound states and composite particles can couple to gauge fields with order-unity strength, so charge neutrality is not a proxy for loop suppression.

Reading between the lines

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

A practical consequence is that global fits of Higgs couplings should treat CP-even and CP-odd field-strength operators as separate free parameters, because the paper's argument removes a theoretical prior that would otherwise shrink some of them.
The ambiguity suggests a sharper UV test: construct two UV completions that match to the same low-energy S-matrix but differ by field redefinitions; if they predict different non-minimal Wilson coefficients, minimal coupling has no observable meaning beyond gauge invariance.
The same logic applies to any proposed ordering of EFT coefficients that is not tied to a symmetry; such an ordering must be preserved by renormalization-group running to be meaningful.
The bound-state examples hint that precision atomic and quarkonium measurements could be used to bound how strongly non-minimal couplings are generated, but the paper does not turn this into a quantitative program.

Original abstract

The principle of minimal coupling has been used in the study of Higgs boson interactions to argue that certain higher dimensional operators in the low-energy effective theory generalization of the Standard Model are suppressed by loop factors, and thus smaller than others. It also has been extensively used to analyze beyond-the-standard-model theories. We show that in field theory, and even in quantum mechanics, the concept of minimal coupling is ill-defined and inapplicable as a general principle, and give many pedagogical examples which illustrate this fact. We also clarify some related misconceptions about the dynamics of strongly coupled gauge theories. Many arguments in the literature on Higgs boson interactions that use minimal coupling, particularly in pseudo-Goldstone Higgs theories, are inherently flawed.

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-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,263 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.12 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.03 · compares with prior results 0.57 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; show every role.

The principle of minimal coupling has been used in the study of Higgs boson interactions to argue that certain higher dimensional operators in the low-energy effective theory generalization of the Standard Model are suppressed by loop factors, and thus smaller than others. It also has been extensively used to analyze beyond-the-standard-model theories. We show that in field theory, and even in quantum mechanics, the concept of minimal coupling is ill-defined and inapplicable as a general principle, and give many pedagogical examples which illustrate this fact. We also clarify some related misconceptions about the dynamics of strongly coupled gauge theories. Many arguments in the literature on Higgs boson interactions that use minimal coupling, particularly in pseudo-Goldstone Higgs theories, are inherently flawed.

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	The principle of minimal coupling has been used in the study of Higgs boson interactions to argue that certain higher dimensional operators in the low-energy effective theory generalization of the Standard Model are suppressed by loop factors, and thus smaller than others.	background 1.00	0.04	0.03	asserted 0.94
s1	It also has been extensively used to analyze beyond-the-standard-model theories.	background 1.00	0.03	0.02	asserted 1.00
s2	We show that in field theory, and even in quantum mechanics, the concept of minimal coupling is ill-defined and inapplicable as a general principle, and give many pedagogical examples which illustrate this fact.	result 0.64	0.95	0.93	asserted 1.00
s3	We also clarify some related misconceptions about the dynamics of strongly coupled gauge theories.	contribution 0.96	0.86	0.04	asserted 0.95
s4	Many arguments in the literature on Higgs boson interactions that use minimal coupling, particularly in pseudo-Goldstone Higgs theories, are inherently flawed.	result 0.75	0.56	0.28	asserted 1.00

What kind of paper the classifier reports this is: position or commentary 0.69 · evidence from argument 0.72.

What was examined, and what was not

sources read

arXiv listing abstract v2

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.** (a) Gauge boson contribution to the pseudo-Goldstone boson mass m2 HH†H (b) Higgs-gauge interaction H†HFµνF µν . the weak gauge coupling. The theory is constructed so that a pseudo-Goldstone multiplet with the quantum numbers of the SM Higgs doublet develops a vacuum expectation value v ΛS, which breaks SU(2) × U(1) down to U(1) electromagnetism. In PGB Higgs models, the leading sigma model interaction terms are L2 = f 2 2 (DµΣ)† (DµΣ) + . . . (6.11) There are also all possible higher dimension operators consistent with the EFT expansion in powers of 1/ΛS. The PGB Higgs theory, just…

**Figure 2.** Meson exchange contribution to π − π scattering. We have explained at length in this paper why minimal coupling operator classification schemes are invalid in EFT. Ref. [13] uses such a scheme to argue that h → γγ is loopsuppressed in the EFT, and that this loop suppression is preserved under RG running. 7 Other Misconceptions In this section, we discuss the invalidity of other arguments which have been advanced in the literature, and are related to the minimal coupling arguments discussed earlier. 7.1 Confining Large N Theories are not like the Linear Sigma Model Strongly…

Figures reproduced from arXiv:1305.0017 by the authors.

References

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Forward citations

Cited by 27 Pith papers

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

Patterns of Strong Coupling for LHC Searches
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A classification of strongly coupled composite scenarios for the Standard Model shows that symmetry-controlled higher-derivative interactions can make dimension-8 operators dominate dimension-6 ones within the effecti...
Renormalization Group Evolution of the Standard Model Dimension Six Operators I: Formalism and lambda Dependence
hep-ph 2013-08 accept novelty 7.0 of 10

The paper computes the Higgs-self-coupling and Yukawa parts of the one-loop anomalous dimension matrix for all 59 Standard Model dimension-six operators, plus their full one-loop effect on Standard Model parameter running.
EWPD in the SMEFT to dimension eight
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Dimension-eight SMEFT corrections to LEP electroweak precision observables are computed and tabulated, showing that neglected higher-order terms can shift fitted Wilson coefficients by amounts comparable to the dimens...
Exact SMEFT formulation and expansion to $\mathcal{O}(v^4/\Lambda^4)$
hep-ph 2020-07 accept novelty 6.0 of 10

Full O(v^4/Lambda^4) SMEFT predictions for h to gamma gamma, h to Z gamma, and Z to fermions differ from the widely used squared-dimension-six shortcut, and the input-parameter scheme dependence grows at higher order.
Effective description of general extensions of the Standard Model: the complete tree-level dictionary
hep-ph 2017-11 accept novelty 6.0 of 10

A complete tree-level dictionary mapping general heavy scalar, fermion and vector extensions of the Standard Model onto the dimension-six Wilson coefficients of the Standard Model effective field theory in the Warsaw basis.

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The SMEFTsim package, theory and tools
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SMEFTsim provides a complete, publicly released FeynRules/UFO implementation of the dimension-six Warsaw-basis SMEFT in three flavour scenarios and two electroweak input schemes, plus a resonance-based scheme for redu...
Analysis of General Power Counting Rules in Effective Field Theory
hep-ph 2016-01 accept novelty 6.0 of 10

General EFT power counting rules in d dimensions show that cross sections are set by Λ counting, not chiral counting, and fix the d-dimensional relation between Λ and the decay constant f.
2, 84, 30, 993, 560, 15456, 11962, 261485, ...: Higher dimension operators in the SM EFT
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The paper provides corrected counts of independent operators in the Standard Model effective field theory at dimensions 7 and 8, and full counts up to dimension 12 for arbitrary fermion generations.
Effective theories of universal theories
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Universal theories, the class for which the S and T oblique parameters are meaningful, are shown to be fully described at dimension six by 16 basis-independent parameters, with explicit dictionaries to four SMEFT bases.
NLO Higgs Effective Field Theory and kappa-framework
hep-ph 2015-05 conditional novelty 6.0 of 10

At next-to-leading order the Higgs kappa-framework is replaced by a dimension-six effective field theory in which each kappa parameter is a fixed linear combination of Wilson coefficients plus a process-dependent non-...
On one-loop corrections in the standard model effective field theory; the $\Gamma(h \rightarrow \gamma \, \gamma)$ case
hep-ph 2015-05 accept novelty 6.0 of 10

The complete one-loop SMEFT contribution to Higgs-to-diphoton from the O_HB, O_HW and O_HWB operators is computed, showing that renormalization-group running alone misses both vacuum-expectation-value running and pure...
Towards consistent Electroweak Precision Data constraints in the SMEFT
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Four-fermion and photon-Z interference terms neglected in past SMEFT fits shift the Z partial widths extracted from LEP data by order Gamma_Z M_Z / v^2, comparable to the percent-level bounds on anomalous Z couplings.
A Systematic Approach to the SILH Lagrangian
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The SILH Lagrangian for a strongly interacting light Higgs is shown to follow as the first-order-in-xi limit of the electroweak chiral Lagrangian, with a consistent power counting and a single custodial-breaking spurion.
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The paper provides a covariant one-loop matching method and a bosonic-sector map from Standard Model effective field theory Wilson coefficients to electroweak and Higgs precision observables.
Renormalization Group Evolution of the Standard Model Dimension Six Operators III: Gauge Coupling Dependence and Phenomenology
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The authors compute the gauge-coupling terms of the one-loop anomalous dimension matrix for the 59 dimension-six SMEFT operators, completing the full 2499 x 2499 matrix.
Renormalization Group Evolution of the Standard Model Dimension Six Operators II: Yukawa Dependence
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The paper tabulates the complete order-y^2 and order-y^4 terms of the one-loop anomalous dimension matrix for the 59 dimension-six operators of the Standard Model effective field theory, completing the gauge-coupling-...
Loops and trees in generic EFTs
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Many dimension-6 contributions to one-loop helicity amplitudes in a generic EFT vanish exactly, rational parts included, suggesting tree-level helicity selection rules persist beyond one loop.
Studies of Dimension-Six EFT effects in Vector Boson Scattering
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A simulation of pp -> ZZjj shows that dimension-six EFT operators beyond the anomalous gauge coupling set change VBS rates and shapes at the ten-percent level, so full-basis fits are needed.
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CoDEx automates tree- and one-loop matching of heavy BSM fields onto dimension-6 SMEFT operators in the Warsaw and SILH bases, with optional RGE running to the electroweak scale.
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Combining LHC Run 1 and Run 2 Higgs and diboson data with precision electroweak measurements constrains 20 dimension-6 SMEFT operators and finds no significant deviation from the Standard Model.
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Projected high-luminosity LHC data on gluon-fusion Higgs pair production into b-bbar tau-tau would constrain the dimension-6 Higgs self-coupling modifier c6 to roughly |c6| < 0.6 at 1 sigma, given projected single-Hig...
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A form factor decomposition of the Higgs-vector-fermion vertex shows that the Higgs-plus-vector-boson invariant-mass spectrum at the LHC can probe momentum-dependent new physics and help distinguish linear from nonlin...
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A one-loop renormalization-group analysis reduces the dimension-six parity- and time-reversal-violating operators that survive to hadronic scales to a set of seven, with explicit coefficients and neutron-EDM bounds.
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A global fit to electroweak precision data with a 126 GeV Higgs constrains the Higgs-to-gauge-boson coupling to within about 5% of the Standard Model value, strongly disfavoring generic composite Higgs models.
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The paper argues that leading-order SMEFT fits to LHC data are insufficient at current and projected precision, and specifies a Warsaw-basis leading-order construction designed to be extended to next-to-leading order.
The Standard Model as an Effective Field Theory
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A pedagogical review arguing that LHC precision data, especially Higgs measurements, should be projected into consistent effective field theories (SMEFT or HEFT) rather than into ad hoc parameterizations.

How the literature uses it

42 arXiv papers cite this in 61 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 45 sentences without a confident label

Contests it 1 paper

As initially formulated in Ref. [ref] , the approach to Meta-matching had several other essential inconsistencies, see discussion in Refs. [ref] .

The Standard Model as an Effective Field Theory · UV matching scenarios · contest 0.84

Uses its method or tool 1 paper

Specifically, we draw random coefficient values according to gaussian distributions with zero mean and root mean square equal to 1 for `tree-level' Wilson coefficients and 0.01 for `loop-level' Wilson coefficients as classified by Refs. [ref] .

EWPD in the SMEFT to dimension eight · SMEFT bottom up · 0.94

Cites it as support 1 paper

The former normalization is not required to respect $SU_C(3)×SU_L(2) ×U_Y(1)$ symmetry and the latter is not model independent[A historically widespread approach of suppressing operators containing field strengths by loop factors was shown to not be a model independent EFT statement in Ref. [ref] .

The SMEFTsim package, theory and tools · Operator normalizations · 0.98

Cites it as background 12 papers

A more detailed discussion of this point, as well as related aspects of minimal coupling are discussed in detail in Ref. [ref] .

Analysis of General Power Counting Rules in Effective Field Theory · Master Formula in $$̣ Dimensions$ · 0.89
For more recent discussions see Refs. [ref]

Studies of Dimension-Six EFT effects in Vector Boson Scattering · Four-fermion operators · 0.98
For strongly interacting BSM, we can use the Higgs effective-field-theory (HEFT) [ref] , in which the electroweak symmetry $SU(2)×U(1)$ is realized nonlinearly.

Scalar and fermion on-shell amplitudes in generalized Higgs effective field theory · Introduction · 0.87

What it cites · 23 sentences without a confident label

Contests 1 paper

which, when expanded out, give contributions to operators that have been argued to be loop suppressed in Ref. [ref] , but without any loop suppression factor.

Renormalization of dimension-six operators relevant for the Higgs decays $h\rightarrow \gamma\gamma,\gamma Z$ · The Higgs as a Pseudo-Goldstone Boson · contest 0.80

Uses data or values from 1 paper

The best fit values of $L_i$ are given in Ref. [ref] .

Review of lattice results concerning low energy particle physics · Chiral Perturbation Theory · 0.95

Cites as background 4 papers

The analysis of Ref. [ref] has been quite influential, and it has led to significant phenomenological work on BSM Higgs boson interactions [ref] .

Theoretical Constraints on the Higgs Effective Couplings · Introduction · 0.83
The analysis of Ref. [ref] has been quite influential, and it has led to significant phenomenological work on BSM Higgs boson interactions [ref] .

Renormalization of dimension-six operators relevant for the Higgs decays $h\rightarrow \gamma\gamma,\gamma Z$ · Introduction · 0.83
The analysis of Ref. [ref] has been quite influential, and it has led to significant phenomenological work on BSM Higgs boson interactions [ref] .

Higgs At Last · Introduction · 0.83

Discussion

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