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

Dimension-Six Terms in the Standard Model Lagrangian

B. Grzadkowski , M. Iskrzynski , M. Misiak , J. Rosiek This is my paper

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
0 comments X

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.

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)
  1. 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.
  2. 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.
  3. 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.

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    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].

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