arxiv: 2604.25134 · v1 · submitted 2026-04-28 · ✦ hep-ph · hep-th

Recognition: unknown

Basis for non-derivative baryon-number-violating operators

Julian Heeck , Brandon B. Le

Authors on Pith no claims yet

Pith reviewed 2026-05-07 16:17 UTC · model grok-4.3

classification ✦ hep-ph hep-th

keywords baryon number violationSMEFTeffective field theoryoperator basisproton decaynon-derivative operatorslepton number

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The pith

A minimal basis for non-derivative baryon-number-violating operators in the Standard Model Effective Field Theory exists up to dimension 11.

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

The paper constructs a compact list of all independent non-derivative operators that break baryon number within the effective theory extension of the Standard Model. These operators describe possible rare processes such as proton decay and neutron-antineutron oscillations that test fundamental symmetries. The authors apply gauge and flavor symmetries along with equations of motion and integration by parts to remove redundant terms, resulting in shorter lists than earlier compilations. They also supply reduced bases for selected dimension-12 cases with specific changes in baryon and lepton numbers. The outcome is a practical starting point for calculating experimental signatures of baryon-number violation.

Core claim

We present a minimal basis for non-derivative baryon-number-violating operators in the Standard Model Effective Field Theory up to mass dimension 11, as well as for the (ΔB,ΔL) = (2,2) and (2,-2) operators at dimension 12. Compared to existing results, our bases generally contain fewer terms and simpler contractions, although we also highlight select cases where a minimal basis is incompatible with simple structures.

What carries the argument

The central mechanism is the classification of operators by their gauge and flavor representations followed by systematic reduction using equations of motion and integration by parts to eliminate dependent terms.

If this is right

Calculations of proton decay rates and similar processes involve fewer independent coefficients.
Experimental limits from baryon-number violation searches translate directly onto a smaller set of Wilson coefficients.
Ultraviolet model building can be checked against the reduced list to see which completions generate allowed operators.
At certain dimensions a fully minimal basis cannot always retain the simplest possible operator structures.

Where Pith is reading between the lines

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

Global fits of effective theory parameters could become more efficient once this basis is implemented in analysis tools.
The cases where minimality conflicts with simple contractions may indicate deeper relations worth exploring in specific models.
The same reduction technique could be applied to derivative operators or to other quantum-number sectors for further simplifications.

Load-bearing premise

The symmetry constraints and reduction rules applied by the authors capture every independent operator without omissions or duplicates.

What would settle it

Finding one additional operator at dimension 9, 10, or 11 that cannot be rewritten using the listed terms, or showing that two operators in the basis are equivalent after reductions, would falsify the minimality claim.

read the original abstract

We present a minimal basis for non-derivative baryon-number-violating operators in the Standard Model Effective Field Theory up to mass dimension 11, as well as for the $(\Delta B,\Delta L) = (2,2)$ and $(2,-2)$ operators at dimension 12. Compared to existing results, our bases generally contain fewer terms and simpler contractions, although we also highlight select cases where a minimal basis is incompatible with simple structures.

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

Heeck and Le deliver a trimmed-down basis for non-derivative B-violating SMEFT operators up to dimension 11 plus a few at 12, with explicit lists that cut the operator count and simplify contractions relative to prior work.

read the letter

The paper's core contribution is an explicit minimal basis for baryon-number-violating operators in the SMEFT that lack derivatives. They cover all such operators through mass dimension 11 and add the (ΔB, ΔL) = (2,2) and (2,-2) cases at dimension 12. The lists are shorter than earlier compilations and use simpler index contractions in most cases, which directly reduces the bookkeeping when matching to UV models or computing rates for proton decay and related processes. They build the basis from gauge and Lorentz invariance plus the required B violation, then systematically remove redundancies using equations of motion and integration by parts. The authors also note a handful of cases where the absolute minimal set does not match the most obvious structures, which shows they are not papering over choices. That honesty is useful because it flags where users might still prefer a slightly larger but cleaner-looking set for calculations. The main soft spot is the completeness of the redundancy removal at these high dimensions. The space of possible contractions grows quickly once flavor and gauge indices are included, so any missed equivalence or overlooked independent operator would affect the claimed minimality. The paper's own mention of conflicting cases makes the enumeration step the part that needs the most scrutiny, and without attached code or fully expanded tables it is hard to double-check by hand. Overall this is a practical tool for anyone doing precision EFT work on B violation. If you calculate proton-decay branching ratios or run operators for baryogenesis bounds, the reduced basis will save steps. It is concrete enough and grounded enough in standard field-theory methods to merit a serious referee, even if the final lists require careful verification during review.

Referee Report

2 major / 2 minor

Summary. The manuscript constructs minimal bases for non-derivative baryon-number-violating operators in the Standard Model Effective Field Theory. It provides complete bases up to mass dimension 11 and extends the treatment to dimension-12 operators with (ΔB, ΔL) = (2,2) and (2,-2). The authors state that the resulting bases contain fewer terms and employ simpler Lorentz and gauge contractions than those appearing in prior literature, while identifying a few cases in which minimality precludes the simplest possible operator structures.

Significance. A compact, verified basis for high-dimensional B-violating operators would reduce the number of independent coefficients that must be tracked in SMEFT analyses of proton decay, neutron-antineutron oscillations, and related processes. If the enumeration is exhaustive and the removal of equations-of-motion and integration-by-parts redundancies is complete, the work supplies a practical tool for both model building and phenomenological studies at dimensions where the operator count grows rapidly.

major comments (2)

[Basis construction and redundancy removal] The central claim of minimality at dimensions 9–12 rests on exhaustive removal of EOM and IBP redundancies after imposing gauge and Lorentz invariance. The manuscript should supply, in the main text or a clearly referenced appendix, an explicit counting of all possible contractions before and after redundancy removal for at least one representative operator class at dimension 11 (e.g., the (ΔB,ΔL)=(1,1) or (1,-1) sector) so that the completeness of the procedure can be verified independently.
[Operator tables (dimensions 9–12)] Table(s) listing the final basis operators must be accompanied by a concise but complete description of the flavor-index contractions and SU(2) representations retained after all redundancies are eliminated. Without this, it is impossible to confirm that no independent operators have been omitted or that equivalent structures have not been retained.

minor comments (2)

[Abstract and introduction] The abstract refers to “select cases where a minimal basis is incompatible with simple structures.” These cases should be identified explicitly in a dedicated paragraph or subsection, with the corresponding operators shown side-by-side in their minimal and non-minimal forms.
[Notation and conventions] Notation for the various SU(2) and Lorentz contractions should be defined once in a single table or paragraph rather than re-introduced each time an operator is written.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful review and constructive suggestions. The comments highlight useful ways to strengthen the verifiability of our basis construction. We address each point below and will incorporate the requested material in a revised version.

read point-by-point responses

Referee: [Basis construction and redundancy removal] The central claim of minimality at dimensions 9–12 rests on exhaustive removal of EOM and IBP redundancies after imposing gauge and Lorentz invariance. The manuscript should supply, in the main text or a clearly referenced appendix, an explicit counting of all possible contractions before and after redundancy removal for at least one representative operator class at dimension 11 (e.g., the (ΔB,ΔL)=(1,1) or (1,-1) sector) so that the completeness of the procedure can be verified independently.

Authors: We agree that an explicit before-and-after count for a representative sector would improve independent verification of the redundancy removal. In the revised manuscript we will add a new appendix that tabulates, for the (ΔB,ΔL)=(1,1) operators at dimension 11, the total number of distinct gauge- and Lorentz-invariant contractions prior to EOM/IBP reduction, the number eliminated by each class of redundancy, and the final count of independent operators. This will be constructed using the same systematic procedure already employed in the paper. revision: yes
Referee: [Operator tables (dimensions 9–12)] Table(s) listing the final basis operators must be accompanied by a concise but complete description of the flavor-index contractions and SU(2) representations retained after all redundancies are eliminated. Without this, it is impossible to confirm that no independent operators have been omitted or that equivalent structures have not been retained.

Authors: We will revise the operator tables (and the accompanying text) to include, for each basis element, a compact notation specifying the retained SU(2) representations and the explicit flavor-index contractions. This will be done without lengthening the tables excessively, by adopting a uniform shorthand already used in the literature for similar SMEFT bases. revision: yes

Circularity Check

0 steps flagged

No circularity: standard enumeration of SMEFT operators via symmetries and redundancy removal

full rationale

The paper constructs its minimal bases by imposing SM gauge and Lorentz invariance plus baryon-number violation, then systematically eliminating operators related by equations of motion and integration by parts. This is a self-contained enumeration relying on well-established field-theory identities rather than any self-definition, fitted parameters renamed as predictions, or load-bearing self-citations. No step reduces the claimed minimality to an input by construction; the result is an explicit listing whose independence can be verified by repeating the same symmetry and redundancy steps.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard assumptions of effective field theory construction; no free parameters or new entities are introduced in the abstract.

axioms (2)

standard math Gauge invariance, Lorentz invariance, and the particle content of the Standard Model
Invoked implicitly as the foundation for operator construction in SMEFT.
domain assumption Restriction to non-derivative operators
Explicitly stated in the title and abstract as the scope of the basis.

pith-pipeline@v0.9.0 · 5359 in / 1197 out tokens · 60410 ms · 2026-05-07T16:17:09.260926+00:00 · methodology

discussion (0)

Reference graph

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