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arxiv: 2512.09287 · v2 · submitted 2025-12-10 · ✦ hep-ph

Nucleon decays into three leptons: contact contributions

Yi Liao , Xiao-Dong Ma , Xiang Zhao This is my paper

Pith reviewed 2026-05-17 00:09 UTC · model grok-4.3

classification ✦ hep-ph
keywords baryon number violationnucleon decaysthree leptonsdimension-9 operatorslow-energy effective theorychiral perturbation theorycontact interactions
0
0 comments X p. Extension

The pith

Dim-9 operators supply contact terms for nucleon decays into three leptons that existing data now tightly bound.

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

This paper investigates baryon-number-violating nucleon decays into three leptons that proceed through direct contact interactions generated by dimension-9 operators in the low-energy effective theory. These modes differ from the more familiar two-body decays and were previously shown to receive only small non-contact contributions from lower-dimensional operators. The authors build the full set of relevant dimension-9 operators, match them to chiral perturbation theory, and compute the resulting decay widths. They then translate current experimental limits on nucleon lifetimes into upper bounds on the operator coefficients. The work also sketches how an ultraviolet-complete model can generate the same operators, opening a path for targeted searches in future neutrino detectors.

Core claim

A complete basis of dimension-9 LEFT operators is constructed that mediate contact contributions to nucleon decays into three leptons. After matching onto chiral perturbation theory, the decay widths are calculated and confronted with existing experimental data, yielding stringent constraints on the operator coefficients.

What carries the argument

Complete basis of dimension-9 LEFT operators for baryon-number-violating three-lepton final states, matched to chiral perturbation theory for matrix-element evaluation.

If this is right

  • Existing data already place strong upper limits on the coefficients of the relevant dim-9 operators.
  • Contact contributions can dominate over the suppressed non-contact terms from dim-6 operators.
  • The same operator framework connects directly to ultraviolet models that could produce observable signals.
  • These decay channels become searchable targets for upcoming neutrino experiments with large fiducial volumes.

Where Pith is reading between the lines

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

  • If the bounds are close to saturation, next-generation detectors could see the first signals in specific three-lepton modes.
  • The operator analysis could be extended to include other baryon-number-violating final states or mixed lepton flavors.
  • Model builders can use the derived coefficients as targets when constructing grand-unified or other high-scale scenarios.
  • The constraints may also inform interpretations of related processes such as neutron-antineutron oscillation searches.

Load-bearing premise

The constructed dimension-9 operator basis is complete for three-lepton final states and the chiral perturbation theory matching captures the leading contributions without large higher-order corrections.

What would settle it

An observed nucleon decay rate into three leptons that exceeds the upper limit set by the derived operator bounds would require either new operators or sizable higher-order corrections.

read the original abstract

Baryon number violating (BNV) nucleon decays into three leptons provide a unique probe of BNV interactions beyond the conventional two-body modes involving a single lepton and a light meson. In a previous work [Nucleon decays into three leptons: noncontact contributions, arXiv:2512.02692.], two of us analyzed the noncontact contributions to these decays arising from dimension-6 (dim-6) operators within the low-energy effective field theory (LEFT), and found that they are severely suppressed due to stringent constraints on these dim-6 operators. In this work, we continue this endeavor by systematically investigating the contact contributions originating from dim-9 LEFT operators. We construct a complete basis of dim-9 operators relevant to these processes, and subsequently match them onto chiral perturbation theory to calculate their decay widths. By employing existing experimental data, we derive stringent constraints on the relevant operators. In addition, we present the analysis of an ultraviolet-complete model to demonstrate its connection with our theoretical framework, thereby facilitating further studies of these exotic nucleon decays in upcoming neutrino experiments with large fiducial masses.

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 paper constructs a complete basis of dimension-9 LEFT operators contributing to baryon-number-violating nucleon decays into three leptons, matches these operators to leading-order chiral perturbation theory, computes the resulting decay widths, and derives constraints on the Wilson coefficients from existing experimental limits. It also presents an ultraviolet-complete model example to illustrate the link to high-scale physics, as a follow-up to the authors' prior analysis of non-contact dim-6 contributions.

Significance. If the operator basis is complete and the chiral matching captures the dominant contributions, the work supplies useful bounds on dim-9 BNV contact interactions that complement two-body decay searches and can inform future neutrino experiments with large fiducial volumes. The explicit UV model connection is a positive feature that aids model-building efforts.

major comments (2)
  1. [§2] §2 (dim-9 operator basis): The manuscript asserts that the enumerated set of dim-9 operators is complete for all B- and L-violating structures that can contribute to N → 3ℓ final states, yet provides no independent cross-check (e.g., Hilbert-series counting or exhaustive listing of Lorentz and flavor structures) against possible missing operators that could interfere after matching. This completeness is load-bearing for the stringency of the derived Wilson-coefficient bounds.
  2. [§3] §3 (ChPT matching): The matching of the dim-9 operators onto leading-order chiral perturbation theory is performed without a quantitative estimate of the size of O(p^4) or higher corrections at the typical momentum transfers of nucleon decays; if these corrections are O(1), they would directly affect the extracted limits.
minor comments (2)
  1. [Table 1] Table 1: the notation for the flavor indices in the operator list is not defined in the caption or surrounding text, making it difficult to reproduce the basis.
  2. [Introduction] The abstract and introduction could more explicitly reference the companion paper (arXiv:2512.02692) when stating how the contact and non-contact contributions are combined.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful review and constructive comments on our manuscript. We appreciate the positive assessment of the work's significance and the explicit UV model connection. Below we provide point-by-point responses to the major comments. We have revised the manuscript accordingly to address the concerns raised.

read point-by-point responses

  1. Referee: [§2] §2 (dim-9 operator basis): The manuscript asserts that the enumerated set of dim-9 operators is complete for all B- and L-violating structures that can contribute to N → 3ℓ final states, yet provides no independent cross-check (e.g., Hilbert-series counting or exhaustive listing of Lorentz and flavor structures) against possible missing operators that could interfere after matching. This completeness is load-bearing for the stringency of the derived Wilson-coefficient bounds.

    Authors: We thank the referee for this important observation. The basis was constructed by enumerating all SM-gauge-invariant three-fermion operators with the required baryon-number violation (ΔB = 1) and lepton-number violation (ΔL = 3) that can produce a nucleon decaying to three leptons, while systematically listing independent Lorentz structures and flavor assignments. Although an explicit Hilbert-series computation was not included in the original draft, the enumeration follows the same procedure used in our prior dim-6 analysis and is consistent with existing literature on higher-dimensional BNV operators. To strengthen the claim, we will add a dedicated paragraph in §2 describing the construction steps in more detail and stating that no additional operators contribute at this dimension. We are happy to incorporate any specific operator the referee believes may have been missed. revision: yes

  2. Referee: [§3] §3 (ChPT matching): The matching of the dim-9 operators onto leading-order chiral perturbation theory is performed without a quantitative estimate of the size of O(p^4) or higher corrections at the typical momentum transfers of nucleon decays; if these corrections are O(1), they would directly affect the extracted limits.

    Authors: We agree that a quantitative assessment of higher-order chiral corrections is desirable. In the present work we employed leading-order matching, which captures the dominant low-energy contribution for these processes. To address the referee’s concern, we will add a brief discussion in §3 using chiral power counting: the typical momentum scale is set by the nucleon mass difference or pion mass (~100–300 MeV), yielding an expansion parameter p/(4πf_π) ≈ 0.2–0.3 and suggesting that NLO corrections are of order 10–30 %. We will explicitly note that the derived Wilson-coefficient bounds carry this theoretical uncertainty and should be interpreted as order-of-magnitude constraints. This addition will be included in the revised manuscript. revision: yes

Circularity Check

0 steps flagged

No significant circularity; constraints derived from external data after internal basis construction and standard matching

full rationale

The paper constructs a complete dim-9 LEFT operator basis for N → 3ℓ contact contributions, matches the operators onto leading-order chiral perturbation theory to obtain decay widths, and then applies existing experimental data to extract bounds on Wilson coefficients. This chain relies on enumeration (performed in the present work), standard EFT matching techniques, and independent external data rather than any fitted parameter renamed as a prediction or any self-definitional loop. The single self-citation to the authors' prior arXiv:2512.02692 paper addresses only the separate non-contact dim-6 contributions and functions as background context, not as a load-bearing premise for the current results. No step reduces by construction to the paper's own inputs, so the derivation remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The analysis rests on standard assumptions of low-energy effective field theory and chiral perturbation theory together with the completeness of the enumerated operator set; no new particles or forces are postulated beyond the operator coefficients.

free parameters (1)
  • Wilson coefficients of dim-9 operators
    The strengths of the new operators remain free parameters that are bounded by experimental data rather than predicted from first principles.
axioms (2)
  • domain assumption The constructed set of dimension-9 operators forms a complete basis for the relevant baryon-number-violating three-lepton processes.
    Invoked when the authors state they build a complete basis before matching to chiral perturbation theory.
  • domain assumption Chiral perturbation theory provides an accurate leading-order description of the nucleon decay matrix elements.
    Used to translate the LEFT operators into decay widths.

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Reference graph

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