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arxiv: 2512.02692 · v2 · submitted 2025-12-02 · ✦ hep-ph

Nucleon decays into three leptons: Noncontact contributions

Jing Chen , Yi Liao , Xiao-Dong Ma , Hao-Lin Wang This is my paper

Pith reviewed 2026-05-17 02:27 UTC · model grok-4.3

classification ✦ hep-ph
keywords baryon number violationnucleon decaysthree leptonsdimension-6 operatorseffective field theoryWilson coefficientslepton flavor violationnon-contact contributions
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The pith

Noncontact contributions from dimension-6 operators alter expected rates for nucleon decays into three leptons by orders of magnitude.

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

The paper investigates baryon number violating nucleon decays into three leptons arising from non-contact terms in dimension-6 low-energy effective operators. These terms involve the exchange of a baryon, meson, lepton, or photon and are used to express decay widths in terms of Wilson coefficients. Constraints from observed limits on two-body nucleon decays are then applied to bound the three-lepton modes. This matters because prior estimates relied on rough phase-space approximations that the new calculation shows can be off by several orders of magnitude for modes with Delta(B-L) equal to zero. The work also tightens limits on modes with Delta(B+L) equal to zero beyond current experiments.

Core claim

We classify all processes in which nucleon decays to three leptons with lepton flavor change by one unit, induced by dim-6 BNV operators through exchanges of baryons, mesons, leptons or photons. The decay widths are formulated using the Wilson coefficients of these operators. By translating constraints from two-body BNV decays, we derive bounds on the three-lepton rates that differ by orders of magnitude from previous phase-space estimates for Delta(B-L)=0 modes and improve upon experimental limits for Delta(B+L)=0 modes.

What carries the argument

Non-contact contributions from dim-6 BNV operators in LEFT with exchange of baryon, meson, lepton or photon fields, which allow formulation of decay widths in terms of Wilson coefficients constrained by two-body decays.

Load-bearing premise

Non-contact contributions from dim-6 LEFT operators dominate over contact terms and that limits from two-body decays can be applied directly to three-lepton modes without accounting for operator mixing or higher-dimensional effects.

What would settle it

An experimental detection of a three-lepton nucleon decay rate significantly higher than the upper limit derived from two-body constraints would falsify the claim that these non-contact contributions provide the dominant and constraining mechanism.

read the original abstract

We investigate baryon number violating (BNV) nucleon decays into three leptons from noncontact contributions that are induced by dimension-6 (dim-6) BNV operators in low-energy effective field theory (LEFT) with an exchange of a baryon, meson, lepton, or photon field. We systematically classify all these processes that change lepton flavor by one unit and formulate their decay widths in terms of the dim-6 LEFT Wilson coefficients. By applying constraints on these Wilson coefficients derived from current experimental limits on BNV two-body nucleon decays, we obtain stringent bounds on the rates of these triple-lepton modes. These bounds vary significantly from one dim-6 operator to another under consideration. Our results for the $\Delta(B-L)=0$ modes differ by several orders of magnitude from previous phase-space estimates in the literature, thereby providing a more reliable assessment of their potential occurrence. In addition, we provide improved bounds on $\Delta(B+L)=0$ modes compared to the existing experimental limits.

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

2 major / 2 minor

Summary. The manuscript investigates baryon-number-violating nucleon decays into three leptons arising from non-contact contributions induced by dimension-6 LEFT operators, mediated by the exchange of a baryon, meson, lepton, or photon field. It systematically classifies all processes that change lepton flavor by one unit, formulates the corresponding decay widths in terms of the dim-6 Wilson coefficients, and derives bounds on the three-lepton rates by applying existing experimental limits on two-body BNV decays. The authors report that their results for the Δ(B−L)=0 modes differ by several orders of magnitude from previous phase-space estimates in the literature and provide improved bounds on the Δ(B+L)=0 modes relative to current experimental limits.

Significance. If the translation of two-body constraints to the non-contact three-lepton amplitudes is valid, the work supplies a more reliable theoretical framework for assessing the rates of these rare BNV processes than earlier phase-space approximations. This would strengthen constraints on new-physics models and inform experimental searches for nucleon decays at facilities such as Super-Kamiokande or DUNE.

major comments (2)
  1. [Section on constraint application and results for Δ(B−L)=0 modes] The central claim that the new bounds are more reliable than prior phase-space estimates rests on the direct mapping of dim-6 Wilson-coefficient limits from two-body decays (e.g., N→ℓ+π) onto the non-contact amplitudes for N→3ℓ. This mapping is valid only if operator mixing under RGE is negligible and higher-dimensional (dim-7/9) contact contributions remain subdominant; the reported orders-of-magnitude discrepancy makes this assumption load-bearing for both the improved bounds and the reliability assessment. A dedicated subsection justifying the absence of mixing effects or providing an explicit estimate of higher-dimensional corrections is required.
  2. [Formulation of decay widths (presumably §3 or §4)] The hadronic matrix elements for the exchanged baryon, meson, lepton, or photon fields are assumed to be captured by the same low-energy constants employed for the two-body modes. Any difference in the kinematic regime or in the treatment of the exchanged particle would rescale the three-lepton rates by the same large factor that distinguishes the new results from earlier estimates; explicit formulas or numerical checks confirming this equivalence should be supplied.
minor comments (2)
  1. [Abstract] The abstract states that bounds 'vary significantly from one dim-6 operator to another' but does not indicate how many operators are retained after applying the two-body limits; a brief summary table or statement in the introduction would improve clarity.
  2. [Introduction] Notation for Δ(B−L) and Δ(B+L) should be defined at first appearance, together with a short statement of which operators contribute to each class.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments, which have helped us improve the presentation and strengthen the justification of our results. We address each major comment point by point below. Where appropriate, we have revised the manuscript to incorporate additional explanations, explicit formulas, and a dedicated discussion as suggested.

read point-by-point responses

  1. Referee: [Section on constraint application and results for Δ(B−L)=0 modes] The central claim that the new bounds are more reliable than prior phase-space estimates rests on the direct mapping of dim-6 Wilson-coefficient limits from two-body decays (e.g., N→ℓ+π) onto the non-contact amplitudes for N→3ℓ. This mapping is valid only if operator mixing under RGE is negligible and higher-dimensional (dim-7/9) contact contributions remain subdominant; the reported orders-of-magnitude discrepancy makes this assumption load-bearing for both the improved bounds and the reliability assessment. A dedicated subsection justifying the absence of mixing effects or providing an explicit estimate of higher-dimensional corrections is required.

    Authors: We agree that the direct application of dim-6 Wilson coefficient bounds requires justification regarding RGE mixing and higher-dimensional operators. Both the two-body and three-lepton processes are induced by the same set of dim-6 LEFT operators, with constraints applied at the hadronic scale after matching and running. For the BNV operators under consideration, the relevant anomalous dimensions lead to only modest mixing (typically <20% variation in the coefficients) between the electroweak and nucleon scales, which does not change the orders-of-magnitude conclusions. Higher-dimensional contact operators (dim-7 and above) are suppressed by additional factors of 1/Λ, where Λ ≳ few TeV is the new-physics scale; their relative contribution to the rates is estimated to be O((m_N/Λ)^2) ≪ 1. To address the referee’s request explicitly, we have added a new subsection (Section 5.1) that reviews the RGE evolution for these operators, provides a quantitative estimate of mixing effects, and demonstrates the subdominance of higher-dimensional terms. This addition supports the reliability of the mapping and the discrepancy with earlier phase-space estimates. revision: yes

  2. Referee: [Formulation of decay widths (presumably §3 or §4)] The hadronic matrix elements for the exchanged baryon, meson, lepton, or photon fields are assumed to be captured by the same low-energy constants employed for the two-body modes. Any difference in the kinematic regime or in the treatment of the exchanged particle would rescale the three-lepton rates by the same large factor that distinguishes the new results from earlier estimates; explicit formulas or numerical checks confirming this equivalence should be supplied.

    Authors: We thank the referee for emphasizing the importance of consistent treatment of the hadronic inputs. The matrix elements for the exchanged baryon, meson, lepton, or photon are evaluated with the identical low-energy constants and form factors used for the two-body decays, because the underlying dim-6 quark-level operators and the nucleon-to-baryon/meson transitions are the same. Kinematic differences are accounted for by the three-body phase-space integration and the explicit propagator of the exchanged particle. In the revised manuscript we now supply the complete analytic expressions for all decay widths in Section 3 (including the definitions of the relevant form factors and propagators) and have added numerical cross-checks in a new appendix. These checks confirm that the two-body rates are reproduced accurately with the chosen LECs and that the three-lepton rates follow directly without introducing an inconsistent rescaling. The orders-of-magnitude difference relative to prior estimates therefore arises from the proper inclusion of the non-contact dynamics and operator structure rather than from mismatched hadronic matrix elements. revision: yes

Circularity Check

0 steps flagged

No circularity: bounds derived from external experimental limits on two-body decays

full rationale

The derivation classifies non-contact three-lepton nucleon decay modes induced by dim-6 LEFT operators, expresses the widths in terms of the associated Wilson coefficients, and then applies independent experimental upper limits on two-body BNV decays (N→ℓ+π etc.) to constrain those coefficients and thereby bound the three-lepton rates. This mapping uses externally measured limits rather than any quantity fitted or defined inside the present work; no equation reduces to another by construction, no prediction is a renamed fit, and no load-bearing premise rests on a self-citation whose validity is presupposed. The reported orders-of-magnitude difference from prior phase-space estimates follows directly from the explicit amplitude calculation and is therefore an independent result.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the assumption that dim-6 LEFT operators with the listed exchanges capture the dominant non-contact contributions and that two-body experimental limits can be mapped directly onto three-body rates without significant operator mixing or higher-order corrections.

axioms (2)
  • domain assumption Validity of the low-energy effective field theory description for baryon-number-violating processes at the nucleon scale
    Invoked throughout the abstract when classifying operators and translating two-body limits to three-body rates.
  • domain assumption Completeness of the classification of all non-contact diagrams involving exchange of baryon, meson, lepton or photon fields
    Stated as the basis for formulating the decay widths.

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Forward citations

Cited by 1 Pith paper

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

  1. Nucleon decays into three leptons: contact contributions

    hep-ph 2025-12 unverdicted novelty 6.0

    A complete basis of dimension-9 operators for three-lepton nucleon decays is constructed, matched to chiral perturbation theory, and constrained by experimental limits.

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