arxiv: 1008.4884 · v3 · submitted 2010-08-28 · ✦ hep-ph

Recognition: 1 theorem link

· Lean Theorem

Dimension-Six Terms in the Standard Model Lagrangian

B. Grzadkowski , M. Iskrzynski , M. Misiak , J. Rosiek

Authors on Pith no claims yet

Pith reviewed 2026-05-14 00:07 UTC · model grok-4.3

classification ✦ hep-ph

keywords dimension-six operatorsStandard Model effective theoryoperator basisbaryon number conservationfield redefinitionseffective field theorynew physics

0 comments

The pith

The Standard Model effective theory contains 59 independent dimension-six operators once redundancies are removed under baryon conservation.

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

This paper reclassifies all dimension-six terms that can appear in the Standard Model Lagrangian when it is treated as a low-energy effective theory. Earlier work had counted 80 such operators, but systematic removal of redundant terms via field redefinitions and equations of motion lowers the total to 59: 15 operators with no fermions, 19 with two fermions, and 25 with four fermions. The reduction holds only when baryon number is conserved; dropping that assumption adds four extra four-fermion operators. The smaller basis matters for any model-independent search for new physics at energies above the electroweak scale.

Core claim

Assuming baryon number conservation, the independent dimension-six operators in the Standard Model Lagrangian number 59, consisting of 15 operators without fermions, 19 with two fermion fields, and 25 with four fermion fields, after removing redundancies via field redefinitions and equations of motion. This updates the earlier enumeration of 80 operators.

What carries the argument

Enumeration of dimension-six operators followed by elimination of redundancies using field redefinitions and equations of motion, performed separately in the zero-, two-, and four-fermion sectors.

If this is right

Precision measurements at colliders and low-energy experiments can be interpreted with a non-redundant set of 59 coefficients rather than 80.
Global fits to new-physics effects become simpler because the operator basis contains no linearly dependent terms.
When baryon-number violation is permitted, the four-fermion sector gains exactly four additional independent operators.
Higher-order calculations in the effective theory can now be performed directly on the reduced set without first removing redundancies by hand.

Where Pith is reading between the lines

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

The same redefinition technique could be applied to dimension-eight operators to produce a compact basis for more precise effective-theory studies.
Existing computer codes that generate or constrain dimension-six operators should be updated to the 59-operator list to avoid double-counting.
Global analyses of electroweak precision data and flavor observables would benefit from refitting with the smaller basis to obtain tighter bounds on the coefficients.
The classification supplies a standard reference that future model builders can use to map ultraviolet completions onto a minimal set of low-energy parameters.

Load-bearing premise

Baryon number is exactly conserved and every redundancy has been correctly removed by the chosen field redefinitions and equations of motion.

What would settle it

An explicit field redefinition or equation of motion that expresses one listed operator as a linear combination of the others would reduce the count below 59 and falsify the basis.

read the original abstract

When the Standard Model is considered as an effective low-energy theory, higher dimensional interaction terms appear in the Lagrangian. Dimension-six terms have been enumerated in the classical article by Buchmueller and Wyler [3]. Although redundance of some of those operators has been already noted in the literature, no updated complete list has been published to date. Here we perform their classification once again from the outset. Assuming baryon number conservation, we find 15 + 19 + 25 = 59 independent operators (barring flavour structure and Hermitian conjugations), as compared to 16 + 35 + 29 = 80 in Ref.[3]. The three summed numbers refer to operators containing 0, 2 and 4 fermion fields. If the assumption of baryon number conservation is relaxed, 4 new operators arise in the four-fermion sector.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper cleans up the dim-6 SM operator list to 59 independent terms by removing redundancies that Buchmueller-Wyler left in.

read the letter

The main takeaway is that Grzadkowski et al. redo the full classification of dimension-six operators in the Standard Model effective theory from scratch. They apply equations of motion and field redefinitions systematically and arrive at 59 independent operators (15 + 19 + 25) under baryon-number conservation, down from the 80 in the 1986 reference. They also list the explicit operators sector by sector, which is the practical part people actually use. If baryon number is not conserved, four extra four-fermion operators appear. That updated count and the concrete lists are what made this the standard reference for SMEFT work at colliders and in precision fits. The paper does the enumeration cleanly and cites the piecemeal redundancy notes that existed before, so the reduction is not just asserted. The method is reproducible in principle: anyone can check the EOM removals and the chosen field redefinitions against the listed operators. Soft spots are limited. The result still sets flavor structure aside, so users have to expand the operators themselves when they need Yukawa or CKM dependence. The baryon-conservation assumption is stated up front and is the usual choice for most applications, but it is an assumption. No numerical cross-checks or sample calculations are included, which is fine for a classification paper but means the count rests on the algebraic steps alone. The citation pattern is straightforward and does not overclaim. This is for anyone who needs the operator basis for EFT calculations or who wants to verify the counting themselves. A reader working on BSM phenomenology or precision electroweak fits will get direct value from the explicit lists. It deserves a serious referee because the central claim is a concrete, verifiable reduction that has held up in the literature since.

Referee Report

0 major / 3 minor

Summary. The manuscript re-derives the complete set of dimension-six operators in the Standard Model effective Lagrangian from first principles. Assuming exact baryon-number conservation, it enumerates 59 independent operators (15 with zero fermions, 19 with two fermions, and 25 with four fermions, excluding flavor structure and Hermitian conjugates) by systematically eliminating redundancies through field redefinitions and equations of motion; this is compared to the 80 operators listed in Buchmueller and Wyler. The paper also notes that dropping the baryon-number assumption adds four additional four-fermion operators.

Significance. If the enumeration and redundancy removal are correct, the work supplies a corrected, explicit operator basis that serves as a standard reference for effective-field-theory analyses of new physics. The sector-by-sector lists allow direct verification and immediate use in phenomenological calculations, reducing the operator count without altering the physical content.

minor comments (3)

The abstract states the final counts but does not indicate the precise field redefinitions or EOM applications used to obtain the reductions (e.g., from 35 to 19 two-fermion operators); a short dedicated paragraph or table in the main text would make the procedure transparent.
The four-fermion sector discussion (when baryon number is relaxed) mentions four new operators but does not list them explicitly; adding the explicit forms would strengthen the claim.
Notation for the operators should be cross-checked against the modern Warsaw-basis conventions to facilitate comparison with subsequent literature.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and for recommending minor revision. The referee's summary correctly captures our re-derivation of the dimension-six SM operators under baryon-number conservation, yielding 59 independent terms (15+19+25) versus the 80 listed by Buchmueller and Wyler.

Circularity Check

0 steps flagged

No significant circularity; direct enumeration from symmetries

full rationale

The paper performs an exhaustive classification of dimension-six SM operators by imposing gauge invariance, Lorentz invariance, and baryon-number conservation, then systematically removing redundancies via field redefinitions and equations of motion. The final count (59 operators) is obtained by explicit listing sector by sector rather than by any fitted parameter, self-referential prediction, or load-bearing self-citation. The comparison to Buchmueller-Wyler is external and does not enter the derivation. No step reduces to its own input by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The classification rests on standard quantum-field-theory axioms plus the explicit assumption of baryon-number conservation; no free parameters or new postulated entities are introduced.

axioms (2)

standard math SM gauge symmetries, Lorentz invariance and canonical dimensions
Used to generate the complete set of dimension-six operators.
domain assumption Baryon number conservation
Explicitly assumed to reduce the four-fermion sector; four additional operators appear when the assumption is dropped.

pith-pipeline@v0.9.0 · 5451 in / 1218 out tokens · 65739 ms · 2026-05-14T00:07:54.221147+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

Foundation/DimensionForcing.lean dimension_forced unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Assuming baryon number conservation, we find 15 + 19 + 25 = 59 independent operators (barring flavour structure and Hermitian conjugations), as compared to 16 + 35 + 29 = 80 in Ref.[3].

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 24 Pith papers

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

Finite-temperature operator basis on $\mathbb{R}^3 \times S^1$ for SMEFT
hep-th 2026-05 unverdicted novelty 8.0

The paper delivers the first complete non-redundant dimension-six operator basis for SMEFT at finite temperature using the Hilbert series on R^3 x S^1.
When Two Loops Matter: Electroweak Precision in the SMEFT
hep-ph 2026-04 unverdicted novelty 7.0

A modification to the top-Higgs Yukawa coupling in SMEFT induces a two-loop shift in the W mass through a large anomalous dimension, providing a new indirect probe via electroweak precision observables.
Basis for non-derivative baryon-number-violating operators
hep-ph 2026-04 unverdicted novelty 7.0

A minimal basis with fewer terms and simpler contractions is given for non-derivative baryon-number-violating operators in SMEFT up to mass dimension 11.
Complete UV Resonances of SMEFT Dim-9 Operators for Short-range Neutrinoless Double Beta Decay
hep-ph 2026-03 unverdicted novelty 7.0

Classification of 440 minimal UV completions for SMEFT dim-9 operators in short-range 0νββ decay, including 324 with vector mediators.
Quantumness of top quark pairs produced at LHC within SMEFT framework
hep-ph 2026-05 unverdicted novelty 6.0

Quantum information observables in LHC top quark pair production are modified by SMEFT dimension-6 operators in ways that depend on their CP properties, offering a complementary probe beyond standard observables.
SMEFT everywhere: a NLO study of $\boldsymbol{pp \to t\bar{t}H}$ with decaying tops
hep-ph 2026-05 unverdicted novelty 6.0

NLO QCD corrections to pp to ttbar H in the di-lepton channel with SMEFT operators included in both production and top-quark decays.
Crossing into the $m_a > f_a$ Region for Leptophilic ALPs
hep-ph 2026-04 unverdicted novelty 6.0

Leptophilic ALPs with m_a > f_a can explain the electron anomalous magnetic moment tension over a large parameter space and are testable via μ→e conversion.
Null Tests and Lepton Universality in $\Xi_{cc}$ Baryon Decays
hep-ph 2026-04 unverdicted novelty 6.0

Doubly charmed baryon decays provide symmetry-protected null tests for nonfactorizable QCD and lepton-universality ratios sensitive to new physics at multi-TeV scales.
EFT Pathways to $|\Delta B| =2$: Chiral Constructions and Phenomenology
hep-ph 2026-04 unverdicted novelty 6.0

A chiral EFT framework is constructed for |ΔB|=2 interactions that matches SMEFT operators to low-energy baryon processes and identifies new dinucleon decay channels sensitive to previously unconstrained operator structures.
Matchotter: An Automated Tool for Dimensional Reduction at Finite Temperature
hep-ph 2026-04 unverdicted novelty 6.0

Matchotter automates one-loop finite-temperature dimensional reduction and supersoft matching for generic Lagrangians using functional techniques.
Probing Higgs and Top Interactions through the Muon Lens at multi-TeV Muon Colliders
hep-ph 2026-04 unverdicted novelty 6.0

A 10 TeV muon collider could improve existing bounds on muon-Higgs-gauge and muon-top interactions by up to an order of magnitude over current limits and FCC-ee projections.
Electroweak Restoration: SMEFT and HEFT
hep-ph 2026-05 unverdicted novelty 5.0

High-energy amplitude ratios for longitudinal di-boson production approach unity in the SM and dimension-6 SMEFT but not necessarily in HEFT, making the W±_L Z_L to W±_L h cross section ratio a promising discriminator...
The EDM inverse problem: Identifying the sources of CP violation and PQ breaking with electric dipole moments
hep-ph 2026-04 unverdicted novelty 5.0

Six classes of CP-violating operators near the QCD scale produce distinct EDM patterns that enable discrimination of their origins and distinguish high-scale versus low-energy sources of the axion vacuum expectation value.
Solving Cosmological Puzzles using Finite Temperature $\nu$SMEFT
hep-ph 2026-04 unverdicted novelty 5.0

A minimal extension of the Standard Model with three heavy Majorana neutrinos simultaneously realizes fermionic dark matter, a strong first-order electroweak phase transition, and low-scale resonant leptogenesis consi...
New Physics Reach through Precision at Future Colliders: a Multi-Pronged Approach
hep-ph 2026-04 unverdicted novelty 5.0

Future e+e- colliders can constrain new physics through precision Higgs and electroweak measurements in Higgs-coupling, EFT, and SMEFT frameworks, with updated SMEFiT code released.
Astrophysical bounds on the high-energy evolution of neutrino mixing
hep-ph 2026-04 unverdicted novelty 5.0

High-energy astrophysical neutrinos can constrain the running of neutrino mixing parameters with energy, with future multi-detector setups forecast to set strong bounds despite astrophysical uncertainties.
Probing $\tau$ lepton dipole moments at future Lepton Colliders
hep-ph 2026-04 unverdicted novelty 5.0

Future lepton colliders can improve existing constraints on the tau lepton's dipole moments by several orders of magnitude through complementary channels.
The Spurion Massive EFT (SMEFT)
hep-ph 2026-04 unverdicted novelty 5.0

SMEFT contact terms are decomposed into spurion structures whose coefficients come from higher-order Higgs singlet combinations, with W/Z masses, mixing, and fermion couplings fully saturated by dimension-eight operators.
CPT Violation, Mirror World and Implications for Baryon Asymmetry
hep-ph 2026-03 unverdicted novelty 5.0

A globally CPT-symmetric paired-universe model with local violations produces inflaton mass splitting that modifies reheating and accounts for observed matter-antimatter asymmetry.
Exploring new resonances with direct top flavor changing interactions
hep-ph 2026-04 unverdicted novelty 4.0

Three new physics resonances with top flavor-changing couplings are mapped to SMEFT operators at the electroweak scale and their phenomenology is analyzed.
Spin effects in the tau-lepton pair induced by anomalous magnetic and electric dipole moments
hep-ph 2026-02 unverdicted novelty 4.0

Anomalous dipole moments of the tau lepton produce measurable effects on tau polarization and spin correlations in gamma-gamma and quark-antiquark induced pair production.
Search for new physics in triple boson production in proton-proton collisions at $\sqrt{s}$ = 13 TeV using the effective field theory approach
hep-ex 2026-05 unverdicted novelty 3.0

No excess in VVV production; bounds set on c_W/Λ² (-0.13 to 0.12 TeV^{-2}) and c_Hq3/Λ² (-0.24 to 0.21 TeV^{-2}) at 95% CL.
Signals of Doomsday III: Cosmological signatures of the late time $U(1)_{EM}$ symmetry breaking
hep-ph 2026-04 unverdicted novelty 3.0

A model of late-time U(1)EM symmetry breaking via scalar-driven first-order phase transition predicts high-energy photon and neutrino bursts as long-range precursors detectable by multi-messenger facilities.
Recent Developments in SMEFT: Theory, Tools, and Phenomenology
hep-ph 2026-04 unverdicted novelty 1.0

A review summarizing recent theory, tools, and phenomenology in the Standard Model Effective Field Theory.

Reference graph

Works this paper leans on

35 extracted references · 35 canonical work pages · cited by 24 Pith papers · 10 internal anchors

[1]

Nakamura et al

K. Nakamura et al. (Particle Data Group) J. Phys. G 37 (2010) 075021, http://pdg.lbl.gov

work page 2010
[2]

Appelquist and J

T. Appelquist and J. Carazzone, Phys. Rev. D 11 (1975) 2856

work page 1975
[3]

Buchmüller and D

W. Buchmüller and D. Wyler, Nucl. Phys. B 268 (1986) 621

work page 1986
[4]

H. D. Politzer, Nucl. Phys. B 172 (1980) 349

work page 1980
[5]

Kluberg-Stern and J

H. Kluberg-Stern and J. B. Zuber, Phys. Rev. D 12 (1975) 3159

work page 1975
[6]

Effective Lagrangians with Higher Order Derivatives

C. Grosse-Knetter, Phys. Rev. D 49 (1994) 6709 [hep-ph/9306321]

work page internal anchor Pith review Pith/arXiv arXiv 1994
[7]

Reduced Effective Lagrangians

C. Arzt, Phys. Lett. B 342 (1995) 189 [hep-ph/9304230]

work page internal anchor Pith review Pith/arXiv arXiv 1995
[8]

Equations of Motion for Effective Lagrangians and Penguins in Rare B-Decays

H. Simma, Z. Phys. C 61 (1994) 67 [hep-ph/9307274]

work page internal anchor Pith review Pith/arXiv arXiv 1994
[9]

Wudka, Int

J. Wudka, Int. J. Mod. Phys. A 9 (1994) 2301 [hep-ph/9406205]

work page arXiv 1994
[10]

Probing Anomalous Top-Quark Couplings Induced by Dim.6 Operators at Photon Colliders

B. Grzadkowski, Z. Hioki, K. Ohkuma and J. Wudka, Nucl. P hys. B 689 (2004) 108 [hep-ph/0310159]

work page internal anchor Pith review Pith/arXiv arXiv 2004
[11]

P. J. Fox et al. , Phys. Rev. D 78 (2008) 054008 [arXiv:0704.1482]

work page internal anchor Pith review Pith/arXiv arXiv 2008
[12]

J. A. Aguilar-Saavedra, Nucl. Phys. B 812 (2009) 181 [arXiv:0811.3842]

work page Pith review arXiv 2009
[13]

J. A. Aguilar-Saavedra, Nucl. Phys. B 821 (2009) 215 [arXiv:0904.2387]

work page internal anchor Pith review Pith/arXiv arXiv 2009
[14]

A. A. Belavin, A. M. Polyakov, A. S. Schwartz and Yu. S. Ty upkin, Phys. Lett. B 59 (1975) 85

work page 1975
[15]

’t Hooft, Phys

G. ’t Hooft, Phys. Rev. D 14 (1976) 3432 [Erratum-ibid. D 18 (1978) 2199]

work page 1976
[16]

Jackiw and C

R. Jackiw and C. Rebbi, Phys. Rev. Lett. 37 (1976) 172

work page 1976
[17]

C. G. . Callan, R. F. Dashen and D. J. Gross, Phys. Lett. B 63 (1976) 334

work page 1976
[18]

Baluni, Phys

V. Baluni, Phys. Rev. D 19 (1979) 2227

work page 1979
[19]

R. J. Crewther, P. Di Vecchia, G. Veneziano and E. Witten , Phys. Lett. B 88 (1979) 123 [Erratum-ibid. B 91 (1980) 487]

work page 1979
[20]

Weinberg, Phys

S. Weinberg, Phys. Rev. Lett. 43 (1979) 1566. 15

work page 1979
[21]

Relativistic Quantum Mechanics

J. D. Bjorken and S. D. Drell, “Relativistic Quantum Mechanics” , McGraw-Hill Inc., 1964

work page 1964
[22]

Buchmüller, B

W. Buchmüller, B. Lampe and N. Vlachos, Phys. Lett. B 197 (1987) 379

work page 1987
[23]

C. Arzt, M. B. Einhorn and J. Wudka, Nucl. Phys. B 433 (1995) 41 [hep-ph/9405214]

work page internal anchor Pith review Pith/arXiv arXiv 1995
[24]

Grzadkowski, Z

B. Grzadkowski, Z. Hioki and M. Szafrański, Phys. Rev. D 58 (1998) 035002 [hep-ph/9712357]

work page arXiv 1998
[25]

Four-Fermi Effective Operators at e^+ e^- ---> \bar{t} t

B. Grzadkowski, Acta Phys. Polon. B 27 (1996) 921 [hep-ph/9511279]

work page internal anchor Pith review Pith/arXiv arXiv 1996
[26]

L. F. Abbott and M. B. Wise, Phys. Rev. D 22 (1980) 2208

work page 1980
[27]

Wilczek and A

F. Wilczek and A. Zee, Phys. Rev. Lett. 43 (1979) 1571

work page 1979
[28]

C. N. Leung, S. T. Love and S. Rao, Z. Phys. C 31 (1986) 433

work page 1986
[29]

Classiﬁcation of higher-dimensional operators in the Sta n- dard Model

M. Iskrzyński, M. Sc. Thesis, “Classiﬁcation of higher-dimensional operators in the Sta n- dard Model” , University of Warsaw, 2010 (in Polish)

work page 2010
[30]

Iskrzyński, talk presented at the 17th IMPRS Worksho p, Munich, July 2010, http://indico.mppmu.mpg.de/indico/conferenceDisplay.py?confId=901

M. Iskrzyński, talk presented at the 17th IMPRS Worksho p, Munich, July 2010, http://indico.mppmu.mpg.de/indico/conferenceDisplay.py?confId=901

work page 2010
[31]

Anomalous Wtb coupling effects in the weak radiative B-meson decay

B. Grzadkowski and M. Misiak, Phys. Rev. D 78 (2008) 077501 [arXiv:0802.1413]

work page internal anchor Pith review Pith/arXiv arXiv 2008
[32]

Grzadkowski and J

B. Grzadkowski and J. Wudka, Phys. Lett. B 364 (1995) 49 [hep-ph/9502415]

work page arXiv 1995
[33]

J. A. Aguilar-Saavedra, arXiv:1008.3562

work page internal anchor Pith review Pith/arXiv arXiv
[34]

Hioki, private communication, 2008

Z. Hioki, private communication, 2008

work page 2008
[35]

Alonso, H

R. Alonso, H. M. Chang, E. E. Jenkins, A. V. Manohar and B. Shotwell, Phys. Lett. B 734 (2014) 302 [arXiv:1405.0486]. 16

work page arXiv 2014