pith. sign in

arxiv: 2605.17525 · v1 · pith:6YOQBNC7new · submitted 2026-05-17 · ✦ hep-ph

Comprehensive investigation of nucleon decays into one lepton plus two mesons

Wei-Qi Fan , Yi Liao , Xiao-Dong Ma This is my paper

Pith reviewed 2026-05-20 12:25 UTC · model grok-4.3

classification ✦ hep-ph
keywords baryon number violationnucleon decaythree-body decayseffective field theorychiral perturbation theoryproton decayWilson coefficientsneutrino modes
0
0 comments X

The pith

Constraints from two-body nucleon decays set much stronger limits on three-body decays to one lepton and two mesons.

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

This paper studies baryon number violating nucleon decays into one lepton and two pseudoscalar mesons inside the low-energy effective field theory. It uses chiral perturbation theory to write the three-body decay rates in terms of the Wilson coefficients of dimension-6 operators. The same operators also drive two-body decays whose experimental limits are already tight, so those limits are imported to bound the three-body channels. The resulting partial-lifetime bounds improve existing experimental limits by orders of magnitude for 31 modes. The derived relations supply a ready-made target list for future nucleon-decay searches.

Core claim

The same dimension-6 LEFT baryon-number-violating operators that induce two-body nucleon decays are matched to three-body final states via chiral perturbation theory; experimental bounds on the two-body modes then constrain the Wilson coefficients and produce improved partial-lifetime limits that are orders of magnitude stronger than present experimental bounds for 22 charged-lepton modes and 9 neutrino modes.

What carries the argument

Dimension-6 baryon-number-violating operators in the low-energy effective field theory whose Wilson coefficients are fixed by two-body data and then used to predict three-body rates through chiral perturbation theory.

If this is right

  • Improved partial-lifetime bounds are obtained for 22 three-body modes containing a charged lepton.
  • Improved partial-lifetime bounds are obtained for 9 three-body modes containing a neutrino or antineutrino.
  • All new bounds lie orders of magnitude above current experimental sensitivities.
  • The operator relations supply concrete targets that future experiments can use to search for these modes.

Where Pith is reading between the lines

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

  • Any future non-observation of three-body modes will automatically tighten the allowed range for the underlying Wilson coefficients already bounded by two-body data.
  • If a three-body signal appears without a matching two-body signal, the assumption that a single set of dimension-6 operators controls both must be revised.
  • The same matching technique can be applied to other multi-body BNV channels once two-body limits are known.

Load-bearing premise

The same dimension-6 operators dominate both two-body and three-body decays and higher-dimensional or unrelated new-physics contributions do not change the three-body rates independently.

What would settle it

Observation of any three-body nucleon decay at a rate that violates the lifetime bound derived from the corresponding two-body constraint.

read the original abstract

We systematically investigate baryon number violating (BNV) nucleon decays into one lepton ($e,\mu,\nu/\bar\nu$) and two pseudoscalar mesons ($\pi\pi,\pi\eta,\pi K$) within the low-energy effective field theory (LEFT) framework. By employing chiral perturbation theory, we obtain general expressions for the decay widths of these three-body nucleon decay modes induced by dimension-6 LEFT BNV operators and express them in terms of the associated Wilson coefficients. Since the same set of LEFT operators contribute to the experimentally well-constrained two-body nucleon decays, we then utilize the experimental bounds on them to constrain the relevant Wilson coefficients. From the obtained constraints, we derive improved limits on the occurrence of 22 three-body modes involving a charged lepton and 9 modes containing a neutrino or an antineutrino, with the new partial lifetime bounds being orders of magnitude stronger than the existing experimental limits. Our framework and derived bounds will facilitate future experimental searches for these nucleon decays.

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 systematically investigates baryon-number-violating nucleon decays into one lepton (e, μ, ν/ν̄) and two pseudoscalar mesons (ππ, πη, πK) in the low-energy effective field theory (LEFT) using dimension-6 BNV operators. Chiral perturbation theory is employed to derive general expressions for the three-body decay widths in terms of the associated Wilson coefficients. Experimental upper bounds on two-body nucleon decays are then used to constrain these coefficients, yielding improved partial-lifetime limits for 22 charged-lepton modes and 9 neutrino modes that are claimed to be orders of magnitude stronger than existing experimental bounds.

Significance. If the central assumptions hold, the work supplies useful new constraints that can guide future experimental searches for these rare processes. The mapping of two-body experimental limits onto three-body rates via shared dimension-6 operators is logically sound and efficiently re-uses existing data. The application of leading-order ChPT to the hadronic matrix elements is a standard and appropriate choice for the low-energy regime. The resulting bounds constitute a concrete, falsifiable input for proton-decay analyses in experiments such as Super-Kamiokande or DUNE.

major comments (2)
  1. [§II] §II (LEFT operator basis and assumptions): The central claim of orders-of-magnitude stronger bounds on the 31 three-body modes rests on the premise that the same dimension-6 LEFT BNV operators dominate both two- and three-body channels. The manuscript does not quantify or bound possible contributions from dimension-7 and higher operators, which could generate additional three-body amplitudes without affecting the two-body constraints used to fix the Wilson coefficients.
  2. [§III] §III (ChPT matrix elements): The leading-order ChPT expressions for the hadronic matrix elements assume dominance by the dimension-6 structures. No estimate is provided for the size of corrections arising from higher-dimensional operators through altered chiral counting or phase-space factors; such corrections would directly loosen the derived partial-lifetime bounds.
minor comments (2)
  1. [Table 1] Table 1 (or equivalent summary table of modes): Explicitly list the 22 charged-lepton and 9 neutrino modes together with the numerical bounds obtained; this would improve readability and allow direct comparison with existing experimental limits.
  2. [Notation] Notation for Wilson coefficients: Ensure uniform labeling (e.g., C_{LL}^{ud}, C_{RR}^{ud}) across the operator definitions, width formulae, and numerical results sections.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and constructive feedback on our manuscript. We appreciate the recognition of the logical soundness of our approach and the utility of the derived bounds. Below, we address each major comment in detail.

read point-by-point responses

  1. Referee: [§II] §II (LEFT operator basis and assumptions): The central claim of orders-of-magnitude stronger bounds on the 31 three-body modes rests on the premise that the same dimension-6 LEFT BNV operators dominate both two- and three-body channels. The manuscript does not quantify or bound possible contributions from dimension-7 and higher operators, which could generate additional three-body amplitudes without affecting the two-body constraints used to fix the Wilson coefficients.

    Authors: We acknowledge that the manuscript focuses on dimension-6 operators without explicitly quantifying the potential impact of higher-dimensional operators. In the context of baryon number violation, dimension-6 operators represent the lowest-dimensional effective operators allowed by the Standard Model gauge symmetries and Lorentz invariance that can induce nucleon decays. Dimension-7 and higher operators are suppressed by additional factors of 1/Λ, where Λ is the scale of new physics, typically expected to be at least a few TeV or higher based on other constraints. To strengthen the manuscript, we will revise Section II to include a brief discussion on the expected suppression of higher-dimensional contributions and note that our bounds assume the dominance of dimension-6 operators, which is standard in such analyses. This will clarify the assumptions and provide a more complete picture. revision: yes

  2. Referee: [§III] §III (ChPT matrix elements): The leading-order ChPT expressions for the hadronic matrix elements assume dominance by the dimension-6 structures. No estimate is provided for the size of corrections arising from higher-dimensional operators through altered chiral counting or phase-space factors; such corrections would directly loosen the derived partial-lifetime bounds.

    Authors: We agree that an estimate of corrections from higher-dimensional operators would be beneficial. The leading-order chiral perturbation theory is employed as it is the standard approach for low-energy hadronic processes involving pseudoscalar mesons, with higher-order terms in the chiral expansion typically suppressed by powers of m_π / (4π f_π) or similar. We will add to Section III a qualitative discussion estimating the size of these corrections, arguing that they are expected to be of order 10-30% based on typical chiral perturbation theory uncertainties, which would not alter the orders-of-magnitude improvement in the bounds. This revision will address the concern about potential loosening of the limits. revision: yes

Circularity Check

0 steps flagged

No circularity: limits derived from independent experimental two-body bounds applied to shared dim-6 operators

full rationale

The paper computes three-body widths via ChPT expressions in terms of Wilson coefficients of dimension-6 LEFT BNV operators, then constrains those coefficients using published experimental upper limits on two-body nucleon decays, and finally evaluates the resulting bounds on three-body partial lifetimes. This chain is self-contained against external benchmarks because the input constraints are independent experimental data rather than fits or definitions internal to the three-body calculation. No self-citations, ansatze, or renamings reduce any central result to the paper's own inputs by construction. The shared-operator assumption is an explicit modeling choice whose validity can be tested externally and does not create definitional circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard assumptions of chiral perturbation theory for low-energy meson-nucleon interactions and on the dominance of dimension-6 operators; no new free parameters are introduced beyond the Wilson coefficients that are constrained by external data.

axioms (1)
  • domain assumption Chiral perturbation theory provides accurate expressions for the matrix elements of the BNV operators between nucleon and meson states at low energies.
    Invoked to obtain general expressions for the three-body decay widths.

pith-pipeline@v0.9.0 · 5701 in / 1245 out tokens · 37467 ms · 2026-05-20T12:25:33.103194+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

25 extracted references · 25 canonical work pages · 6 internal anchors

  1. [1]

    A. D. Sakharov,Violation of CP Invariance, C asymmetry, and baryon asymmetry of the universe,Pisma Zh. Eksp. Teor. Fiz.5(1967) 32–35

    work page 1967

  2. [2]

    Barbieri, D

    R. Barbieri, D. V. Nanopoulos, G. Morchio and F. Strocchi,Neutrino Masses in Grand Unified Theories,Phys. Lett. B90(1980) 91–97

    work page 1980

  3. [3]

    K. S. Babu and R. N. Mohapatra,Predictive neutrino spectrum in minimal SO(10) grand unification,Phys. Rev. Lett.70(1993) 2845–2848, [hep-ph/9209215]

    work page internal anchor Pith review Pith/arXiv arXiv 1993

  4. [4]

    Dark Matter Interpretation of the Neutron Decay Anomaly

    B. Fornal and B. Grinstein,Dark Matter Interpretation of the Neutron Decay Anomaly, Phys. Rev. Lett.120(2018) 191801, [1801.01124]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  5. [5]

    G. Elor, M. Escudero and A. Nelson,Baryogenesis and Dark Matter fromBMesons,Phys. Rev. D99(2019) 035031, [1810.00880]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  6. [6]

    Ge and X.-D

    S.-F. Ge and X.-D. Ma,Nucleon consumption and mass-energy conversion induced by dark matter,Phys. Rev. D110(2024) 115004, [2406.00445]

    work page arXiv 2024

  7. [7]

    L. J. Broussard et al.,Baryon number violation: from nuclear matrix elements to BSM physics,J. Phys. G52(2025) 083001, [2504.16983]. [8]Particle Data Groupcollaboration, S. Navas et al.,Review of particle physics,Phys. Rev. D110(2024) 030001. [9]JUNOcollaboration, F. An et al.,Neutrino Physics with JUNO,J. Phys. G43(2016) 030401, [1507.05613]. [10]Hyper-K...

    work page arXiv 2025

  8. [8]

    M. B. Wise, R. Blankenbecler and L. F. Abbott,Three-body Decays of the Proton,Phys. Rev. D23(1981) 1591

    work page 1981

  9. [9]

    Search for proton decay via $p \to e^{+}\pi^{0}\pi^{0}$ and $p \to \mu^{+}\pi^{0}\pi^{0}$ in 0.401 megaton-years exposure of Super-Kamiokande I-V

    C. McGrew et al.,Search for nucleon decay using the IMB-3 detector,Phys. Rev. D59 (1999) 052004. [16]Kamiokandecollaboration, K. Abe et al.,Search for proton decay viap→e +π0π0 and p→µ +π0π0 in 0.401 megaton-years exposure of Super-Kamiokande I-V,2604.10975

    work page internal anchor Pith review Pith/arXiv arXiv 1999

  10. [10]

    Learned, F

    J. Learned, F. Reines and A. Soni,Limits on Nonconservation of Baryon Number,Phys. Rev. Lett.43(1979) 907

    work page 1979

  11. [11]

    M. L. Cherry, M. Deakyne, K. Lande, C. K. Lee, R. I. Steinberg and B. T. Cleveland, Experimental Test of Baryon Conservation: A New Limit on the Nucleon Lifetime,Phys. Rev. Lett.47(1981) 1507–1510. [19]Frejuscollaboration, C. Berger et al.,Results from the Frejus experiment on nucleon decay modes with charged leptons,Z. Phys. C50(1991) 385–394

    work page 1981

  12. [12]

    E. E. Jenkins, A. V. Manohar and P. Stoffer,Low-Energy Effective Field Theory below the Electroweak Scale: Operators and Matching,JHEP03(2018) 016, [1709.04486]

    work page arXiv 2018

  13. [13]

    Liao, X.-D

    Y. Liao, X.-D. Ma and Q.-Y. Wang,Extending low energy effective field theory with a complete set of dimension-7 operators,JHEP08(2020) 162, [2005.08013]

    work page arXiv 2020

  14. [14]

    Claudson, M

    M. Claudson, M. B. Wise and L. J. Hall,Chiral Lagrangian for Deep Mine Physics,Nucl. Phys. B195(1982) 297–307

    work page 1982

  15. [15]

    Liao, X.-D

    Y. Liao, X.-D. Ma and H.-L. Wang,New Chiral Structures for Baryon Number Violating Nucleon Decays,Phys. Rev. Lett.135(2025) 161801, [2504.14855]

    work page arXiv 2025

  16. [16]

    Liao, X.-D

    Y. Liao, X.-D. Ma and H.-L. Wang,Chiral perturbation theory for baryon-number-violating nucleon decay into a vector meson,Phys. Rev. D112(2025) L031704, [2506.05052]

    work page arXiv 2025

  17. [17]

    W.-Q. Fan, Y. Liao and X.-D. Ma,Nucleon decays into one lepton plus two non-strange mesons,2604.24613

    work page internal anchor Pith review Pith/arXiv arXiv

  18. [18]

    W.-Q. Fan, Y. Liao, X.-D. Ma and H.-L. Wang,Baryon number violating hydrogen decay, Phys. Lett. B862(2025) 139335, [2412.20774]

    work page arXiv 2025

  19. [19]

    J.-S. Yoo, Y. Aoki, P. Boyle, T. Izubuchi, A. Soni and S. Syritsyn,Proton decay matrix elements on the lattice at physical pion mass,Phys. Rev. D105(2022) 074501, [2111.01608]

    work page arXiv 2022

  20. [20]

    E. E. Jenkins and A. V. Manohar,Baryon chiral perturbation theory using a heavy fermion Lagrangian,Phys. Lett. B255(1991) 558–562

    work page 1991

  21. [21]

    Bijnens, H

    J. Bijnens, H. Sonoda and M. B. Wise,On the Validity of Chiral Perturbation Theory for Weak Hyperon Decays,Nucl. Phys. B261(1985) 185–198. – 37 – [30]RQCDcollaboration, G. S. Bali, S. Collins, W. S¨ oldner and S. Weish¨ aupl,Leading order mesonic and baryonic SU(3) low energy constants from Nf=3 lattice QCD,Phys. Rev. D105 (2022) 054516, [2201.05591]. [31...

    work page arXiv 1985

  22. [22]

    Ecker, J

    G. Ecker, J. Gasser, H. Leutwyler, A. Pich and E. de Rafael,Chiral Lagrangians for Massive Spin 1 Fields,Phys. Lett. B223(1989) 425–432

    work page 1989

  23. [23]

    Y. Unal, A. Kucukarslan and S. Scherer,Interaction of the vector-meson octet with the baryon octet in effective field theory,Phys. Rev. C92(2015) 055208, [1510.02205]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  24. [24]

    Kawarabayashi and M

    K. Kawarabayashi and M. Suzuki,Partially conserved axial vector current and the decays of vector mesons,Phys. Rev. Lett.16(1966) 255

    work page 1966

  25. [25]

    Rev.147(1966) 1071–1073

    Riazuddin and Fayyazuddin,Algebra of current components and decay widths of rho and K* mesons,Phys. Rev.147(1966) 1071–1073. – 38 –

    work page 1966