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This paper constructs a baryon- and lepton-number-violating model in which proton decay is suppressed by a dimension-nine operator, so the new scalar masses sit at ~1 TeV and can be produced at the LHC.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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A TeV-scale baryon number violation model with three new scalars evades proton decay limits and predicts a same-sign dimuon plus anti-top signature at the LHC.

T0 review reviewed 2026-08-05 challenge →

load-bearing objection Interesting TeV-scale BNV model with a clean LHC signature, but the central proton-decay scale claim is built on a dimensionally inconsistent formula in Eq. (8) that needs to be fixed before the TeV window is credible. the 5 major comments →

arxiv 2508.21064 v1 pith:5O4KZPIN submitted 2025-08-28 hep-ph hep-ex

A Baryon and Lepton Number Violation Model Testable at the LHC

Amit Bhoonah , Francis Burk , Da Liu , Tong Ou , Deepak Sathyan This is my paper
classification hep-ph hep-ex
keywords baryon number violationlepton number violationproton decaydimension-nine operatorB − L/3 conservationleptoquarkdiquarksame-sign dilepton search
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The reading

Proton decay experiments normally force baryon-number-violating physics up to ~10^15 GeV, out of collider reach. This paper claims that a specific symmetry—conserving B − L/3, not the usual B − L—raises the leading baryon-number-violating interaction to dimension nine, so proton decay is suppressed by five powers of the new-physics mass scale. With that suppression, the Super-Kamiokande limit on p → e+/μ+ νν leaves the new scalar masses around 1 TeV. The paper then shows these scalars give a distinctive LHC final state—an anti-top quark, two same-sign muons, and large missing energy—and constructs a low-scale baryogenesis mechanism from the same interaction. The point of the paper is that baryon number violation can be a TeV-scale, experimentally accessible story rather than a GUT-scale one.

Core claim

The authors construct an explicit renormalizable model in which baryon number changes by −1 and lepton number by −3 while B − L/3 is conserved. Three scalar fields are added: a diquark (two right-handed up-type quarks), a leptoquark (left-handed quark plus lepton), and a singly charged dilepton singlet (two left-handed leptons), with quantum numbers chosen so that none of the Yukawa couplings breaks the global symmetry. Baryon number violation enters only through one soft trilinear scalar coupling, Λ_BNV χ†_LQ χ_uu χ_LL. After integrating out the scalars, the lowest BNV effective operator is dimension nine, O_9 = c_9/M^5_BNV Q^c_iβ L_jα u^c_k P_R u_l L^c_mγ L_nδ ε_αβ ε_γδ, with M^5_BNV = Λ_B

What carries the argument

The load-bearing object is the conserved combination B − L/3, enforced by the scalar quantum numbers, together with the single soft trilinear coupling Λ_BNV χ†_LQ χ_uu χ_LL that violates B and L while preserving it. This forces the lowest BNV effective operator to be dimension nine, O_9, so proton decay is suppressed by five powers of the scalar mass scale instead of the two powers typical of a GUT-scale dimension-six operator. In the proton decay calculation, the controlling mechanism is the two chirality flips: the loop must pass through a charm or top quark at the diquark vertex, kinematically forbidding tree-level decay, and through the constituent up-quark mass, with the two W-loop diag

Load-bearing premise

The TeV window collapses if any interaction other than the single soft trilinear scalar coupling violates baryon number: the argument assumes B − L/3 is exact everywhere except for that one term, so any extra BNV interaction of dimension lower than nine—from the scalar singlet, the neutrino-mass sector, or a UV completion—would make proton decay far faster than predicted.

What would settle it

Super-Kamiokande or Hyper-Kamiokande observing the decay p → e+ νν at a lifetime of about 10^32 years or below would falsify the central scenario: for scalar masses above 1 TeV the model predicts lifetimes above the current 1.7 × 10^32-year bound, typically far beyond 10^35 years at the benchmarks, so any detected event in that channel at present sensitivity rules out the TeV mass window.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • The LHC becomes a baryon-number-violation discovery machine: the uc → χ†_uu → χ†_LQ χ_LL → tbar μ+ μ+ νbar_tau chain gives a clean same-sign dimuon plus boosted anti-top signature with large missing transverse energy.
  • Future proton decay experiments are not the most sensitive probe of this model: for most allowed parameter space the predicted proton lifetimes exceed 10^35 years, while the predicted collider cross-sections are testable now and at the HL-LHC.
  • For the benchmark with m_χuu = 3.5–5.5 TeV, m_χLQ = m_χLL = 1.5 TeV and Λ_BNV = 10 TeV, the signal survives an E_T^miss > 500 GeV cut while the dominant tbar-tW background drops by roughly two orders of magnitude, placing hundreds of signal events within HL-LHC reach.
  • The same BNV interaction can drive baryogenesis: the decay of a ~0.8–1 TeV real scalar singlet through the six-body BNV channel happens far below the electroweak scale, avoiding sphaleron washout, and its CP asymmetry derives from CKM and PMNS phases, so future neutrino experiments can constrain it.
  • Lepton-flavour-violation constraints, particularly μ → eγ, fix the viable couplings: electron couplings of the leptoquark and dilepton must be tiny, while top-muon and top-tau couplings can be O(1), shaping which final states are observable.

Where Pith is reading between the lines

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

  • A broader lesson is that the B − L/3 shortcut is general: any model whose leading BNV operator is dimension nine should share the same TeV mass window and the same same-sign-dilepton plus heavy-flavour search logic, so the proposed search could serve as a template for an entire operator class.
  • The model as presented does not generate neutrino masses, since it violates lepton number by three units while Majorana masses need ΔL = 2. Adding the Zee-Babu-style scalar sector the authors mention will introduce new BNV couplings and new proton decay modes; until that sector is analysed, the exact TeV window should be read as conditional.
  • If the baryogenesis mechanism's Φ also produces a pseudo-Goldstone boson carrying B and L, as the authors speculate, that object could be dark matter or contribute to radiation; its cosmological imprints would provide an independent, testable consequence beyond the collider signature.
  • The LHC channel can certify ΔB = −1 through the anti-top, but the missing neutrino leaves ΔL ambiguous (−1 or −3); given the authors' estimate that the W* channel that would fix ΔL = −3 has negligible rate at the HL-LHC, a future lepton collider may be the only way to directly confirm the claimed ΔL = −3 structure.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

5 major / 5 minor

Summary. The manuscript constructs a renormalizable BSM model with three TeV-scale scalars (a diquark χ_uu, a leptoquark χ_LQ, and a singly charged dilepton χ_LL) and one soft trilinear BNV interaction that conserves B−L/3, giving ΔB=-1 and ΔL=-3. The leading BNV operator is claimed to be dimension nine, and the paper argues that the Super-Kamiokande limit on p→e^+/μ^+νν forces only M_BNV ≳ 1 TeV, leaving the new scalars within LHC reach. Existing low-energy constraints (μ→eγ, LFU, D-meson mixing, dijet searches) are used to fix benchmark couplings. For two diquark masses, a parton-level MadGraph simulation of uc→χ†_uu→χ†_LQχ_LL→tbar μ^+ μ^+ νbar_τ is presented, with ttW as the dominant background after an ETmiss cut. A post-sphaleron baryogenesis mechanism based on decay of a complex singlet Φ is also outlined. The central quantitative claim is the O(TeV) proton-decay bound that opens the LHC window.

Significance. If the TeV bound survives a corrected operator normalization, this is a useful and falsifiable counterexample to the common expectation that proton decay forces BNV resonances to the GUT scale. The proton-decay amplitude is worked out in unusual detail, including the destructive interference between the two loop diagrams, the chirality-flip structure, and the lattice matrix-element input, and the proposed tbar μ^+μ^+ + ETmiss signature with anti-top charge identification is distinctive and worth an experimental look. The baryogenesis connection is a nice bonus, although the quantitative estimate is only order-of-magnitude. The manuscript does not include code, but Appendix B is sufficiently explicit to reproduce the decay-rate calculation. The confidence in the central claim is currently limited by the issues below, not by the physics idea itself.

major comments (5)
  1. [II, Eq. (8)] Eq. (8) is dimensionally inconsistent. With the soft trilinear coupling of Eq. (4), [Λ_BNV]=1, so the right-hand side of M^5_BNV = Λ_BNV(m_χLQ m_χuu m_χLL)^2 has mass dimension 7, not 5. Tree-level exchange of one χ_LQ, one χ_uu and one χ_LL gives coefficient Λ_BNV/(m^2_χLQ m^2_χuu m^2_χLL), i.e. M^5_BNV = m^2_χLQ m^2_χuu m^2_χLL/Λ_BNV, which has the opposite Λ dependence from Eq. (8). For the BP1 masses (m_LQ=m_LL=1.5 TeV, m_uu=3.5 TeV, Λ=10 TeV), the two definitions differ by about a factor 2.5 in M_BNV, and since the proton rate scales roughly as M^-10, the corresponding lifetime changes by about four orders of magnitude. Because Eq. (8) is the stated basis of the paper's 1 TeV bound, the central claim is not yet supported. If Appendix B uses the correct normalization (the overall Λ/m^2_LL prefactor in Eq. (B9) suggests it may), this should be stated explicitly and Eq. (8) corrected o
  2. [III B, Eq. (11) and Eq. (14)] The benchmark Yukawa matrices in Eq. (14) violate the paper's own D0–D0bar mixing bound. Eq. (11) reads |y^ut_uu y^ct_uu| (m_χuu/TeV) ≤ 1.5×10^-2. The benchmark takes |y^ut_uu|=|y^ct_uu|=0.12. For m_χuu=3.5 TeV this gives 0.0504, exceeding the bound by a factor 3.4, and for m_χuu=5.5 TeV by a factor 5.3. These couplings enter the proton-decay amplitude in Eqs. (B9)/(B11) and the diquark width, so the numerical results in Figs. 2, 4, 5 and Table IV are based on a parameter point that is inconsistent with a constraint listed in the same section. The benchmark should be revised (e.g., |y^ut_uu|=|y^ct_uu| ≲ 0.0043 at m_χuu=3.5 TeV) and the proton-lifetime and collider numbers recomputed.
  3. [IV, Table IV and Fig. 4] The collider projection is parton-level and leading order. The signal cross sections in Table IV and the apparent significance in Fig. 4 are obtained without detector simulation, object reconstruction efficiencies, or systematic uncertainties, and the background estimate considers only ttW plus a qualitative WZj comment. The same-sign dilepton searches cited in the text also need control of ttbar, WZ, double-parton and fake/non-prompt lepton backgrounds; the statement that a pT>25 GeV cut suppresses non-prompt leptons is not a quantitative estimate. For a paper whose headline is LHC testability, this is not yet a demonstration of a discovery-level signal. Please either upgrade the background/fake estimate and add systematic uncertainties, or reframe the claim as a strong motivation for a dedicated experimental search.
  4. [III A and Appendix B] The proton-decay bound depends on two hadronic inputs that are not quantified: the constituent quark mass m*_u ≃ m_P/3 and the lattice matrix element α_QCD = 0.014 GeV^3. The amplitude is linear in m*_u, so τ_p scales as 1/m*_u^2; varying m*_u between 150 and 450 MeV changes the lifetime by an order of magnitude, which is hard to reconcile with the footnote's statement that the precise value has no appreciable effect. The loop integrals and phase-space factors are computed with a series of approximations whose combined uncertainty is not assigned. Before the 'roughly M_BNV > 1 TeV' limit can be used as a quantitative result, the authors should provide an error estimate for these inputs and show how the bound shifts.
  5. [II and V] The entire TeV window relies on the assumption that the only BNV interaction at low energies is the dimension-nine operator generated by Eq. (4). Section II motivates Λ_BNV through a complex singlet Φ with VEV, then adds a (Φ*Φ)Φ term to avoid a massless Goldstone. This explicit breaking of the global B−L/3 symmetry is not shown to be harmless: it could generate additional BNV operators, possibly of lower dimension, with coefficients that are not controlled. The baryogenesis section uses the Φr decay through the same BNV vertex, so the UV completion is not just a decorative comment. Please state explicitly that exact B−L/3 with a single soft breaking is an assumption of the model, and, if the Φ completion is part of the proposal, estimate the size of the additional BNV operators it induces.
minor comments (5)
  1. [Figure 7 caption] The caption says 'We always evaluate the perturbative unitarity constraints at √s=√4.5m but show the constraints for √s=√4.5m to show the effect of increasing s.' The second value should be √15m; as written the sentence is self-contradictory.
  2. [V] The text says m_Φr can be O(100) GeV, but the subsequent constraint for decay before the QCD phase transition gives m_Φr > 790 GeV. These statements should be reconciled (e.g., 'O(100 GeV) to O(1 TeV)' or a corrected numerical range).
  3. [III B, Eq. (10)] Eq. (10) is typeset in a garbled way: the factor 1/(m^2_χLQ G_F)^2 and the bracket containing the two amplitude contributions are difficult to parse, and the parenthesis structure is unclear. Please rewrite this expression with unambiguous brackets and define Q_LQ/Q_LL before use.
  4. [III B, footnote 4] The phrase 'we choose not to chase that particular ambulance' is informal and does not belong in a journal report; consider replacing it with a standard statement about not relying on tuned cancellations.
  5. [Appendix B, Eqs. (B9)/(B11)] The symbol α_QCD is used for the lattice matrix element 0.014 GeV^3, which conflicts with the standard meaning of the strong coupling constant; rename it (e.g., α_p) and define it at first use. Similarly, λ*⟨ΛB⟩ should be written as Λ_BNV with its mass dimension made explicit.

Circularity Check

0 steps flagged

No significant circularity: the central proton decay and collider predictions are computed from the model Lagrangian and external inputs, not fitted to the claimed outputs.

full rationale

The paper's central claim is that the proton decay bound in this model is roughly M_BNV > 1 TeV. That bound is obtained by an explicit loop calculation in Appendix B starting from the Yukawa interactions in Eqs. (1)-(3) and the trilinear BNV term in Eq. (4), using lattice matrix elements from Ref. [69] and the constituent quark mass approximation. The proton lifetime is therefore derived, not fitted to the Super-K limit. The benchmark parameter choices are made to saturate other baryon-number-conserving constraints (Tables II and III), and the same parameters are then used to compute LHC signal cross-sections; this is ordinary parameter selection, not circular. No references to the present authors' own prior work are used as load-bearing evidence; the cited dimension-counting results [17,18], the baryogenesis rate formulas [22,23], and the lattice inputs [69] are all external. The possible dimensional inconsistency in Eq. (8) noted by the skeptic is a correctness risk, not a circularity: it does not reduce any prediction to an input. Hence there is no self-definitional step, no fitted quantity renamed as a prediction, and no self-citation chain forcing the result.

Axiom & Free-Parameter Ledger

8 free parameters · 6 axioms · 4 invented entities

The paper introduces three new scalars and one complex singlet, with a set of Yukawa couplings chosen to satisfy existing flavor and collider constraints. The proton lifetime and collider cross-sections are predictions of this chosen benchmark, not fitted to data. The main model-building input is the exact B - 1/3 L symmetry, which is a postulate.

free parameters (8)
  • y_uu (diquark Yukawa matrix) = yuc=0.25, yut=yct=0.12
    Benchmark chosen to saturate dijet and D0 mixing constraints; controls diquark production and decay.
  • y_LL (dilepton Yukawa matrix) = yeμ=0.125, yμτ=0.325, yeτ=10^-3
    Benchmark chosen to saturate lepton flavor universality and μ→eγ bounds; controls dilepton decays.
  • y_LQ (leptoquark Yukawa matrix) = yeu=0.397, yμc=0.949, yτt=1, yμt=1, others 0
    Benchmark chosen to satisfy rare meson decay limits and allow TeV masses; controls leptoquark decays to top and muon.
  • Λ_BNV = 10 TeV
    Trilinear scalar coupling; chosen to satisfy proton lifetime and perturbative unitarity while giving a sizeable BNV branching fraction.
  • m_χuu = 3.5 TeV (BP1), 5.5 TeV (BP2)
    Diquark mass benchmarks chosen for LHC reach; allowed by dijet constraints.
  • m_χLQ = 1.5 TeV
    Leptoquark mass benchmark, set equal to m_χLL.
  • m_χLL = 1.5 TeV
    Dilepton singlet mass benchmark.
  • m_u* (constituent quark mass) = m_P/3 ~ 313 MeV
    Used for the external up quark chirality flip in the proton decay rate; an input to the matrix element, not a Lagrangian parameter.
axioms (6)
  • ad hoc to paper The global symmetry B - 1/3 L is exact in the renormalizable Lagrangian and broken only by the soft trilinear Λ_BNV.
    This is the central model-building assumption; it ensures the leading BNV operator is dimension nine. Section II, Table I and eq. (4).
  • domain assumption The new scalars do not acquire VEVs, with positive quartic couplings.
    Stability of the potential requires λjj>0, λjk>0, and λjH > -μ^2/v^2. Section II.
  • ad hoc to paper The dimension-nine operator is the only low-energy BNV operator.
    Derived from the model fields; no lower-dimensional BNV is generated because each Yukawa interaction conserves B and L.
  • domain assumption Lattice nucleon matrix element parameters α=-β=-0.014 GeV^3 from Ref. [69].
    Used in the proton decay amplitude, eq. (B7).
  • domain assumption Constituent quark mass m_u* = m_P/3 for the valence up quark.
    Provides the second chirality flip in the proton decay loop. Section III.A and eqs. (B8)-(B10).
  • standard math Standard Model gauge symmetry and CKM/PMNS unitarity.
    Used throughout for flavor mixing and loop functions.
invented entities (4)
  • χuu (diquark scalar) independent evidence
    purpose: Couples two right-handed up-type quarks and produces the diquark resonance in the LHC signal.
    Has a specific production channel uc→χuu and a predicted decay chain, with cross-sections that could be searched for.
  • χLQ (leptoquark scalar) independent evidence
    purpose: Couples a left-handed quark and lepton; mediates the collider decay and participates in proton decay.
    Can be produced via the diquark cascade and has existing LHC searches; the model predicts specific top-muon decays.
  • χLL (dilepton singlet scalar) independent evidence
    purpose: Couples two left-handed leptons and produces the same-sign muons in the signal.
    Can be produced in the diquark cascade; predicted decay to muon plus neutrino.
  • Φ (complex scalar singlet) no independent evidence
    purpose: Generates Λ_BNV through its VEV; its real component decays to produce the baryon asymmetry.
    Introduced for baryogenesis; no direct experimental handle beyond the BNV coupling, and the pseudo-Goldstone dark matter is only mentioned as a possibility.

reviewed 2026-08-05 · how reviews work

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Cite this review

Pith. "Pith review of A Baryon and Lepton Number Violation Model Testable at the LHC." pith.science (2026). https://pith.science/paper/5O4KZPIN

@misc{pith2026250821064,
  author       = {Pith},
  title        = {Pith review of: A Baryon and Lepton Number Violation Model Testable at the LHC},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5O4KZPIN}},
  note         = {Machine review of arXiv:2508.21064}
}
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abstract

Proton decay experiments typically constrain baryon number violation to the scale of grand unified theories. From a phenomenological point of view, this makes direct probing of the associated new resonances, such as the X and Y bosons, out of reach for even the most optimistic future experiments. It has, however, been known that certain specific patterns of baryon and lepton number violation can suppress proton decay by multiple powers of the masses of the heavy resonances involved, opening the possibility that the observed limits on the proton lifetime are consistent with baryon number violating physics at energy scales much lower than that of grand unification. We construct an explicit example of such a model which violates baryon number by one unit, $\Delta \text{B} = -1$, and lepton number by three units, $\Delta \text{L} = -3$, and show that despite stringent limits on the predicted $p \rightarrow e^{+}/\mu^{+} \overline{\nu}\overline{\nu}$ mode from the Super-Kamiokande experiment, the masses of the newly introduced elementary particles can be $\mathcal{O}$(TeV). We identify interesting unique signatures of baryon number violation of this model that can be probed both with currently available LHC data and with the upcoming High-Luminosity LHC. We also present a scenario for low-scale baryogenesis within the framework of this model.

Figures

Figures reproduced from arXiv: 2508.21064 by Amit Bhoonah, Da Liu, Deepak Sathyan, Francis Burk, Tong Ou.

Figure 1
Figure 1. Figure 1: FIG. 1. Feynman diagrams for loop level proton decay, where [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Parton level description of the distinctive hadron col [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Density plot (blue) and constraint (solid black line) [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Missing transverse energy distributions for signal and [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Parameter space showing existing constraints (blue), [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Tree and one-loop diagrams of [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Perturbative unitarity constraints on [PITH_FULL_IMAGE:figures/full_fig_p013_7.png] view at source ↗

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Forward citations

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

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    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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This paper was first reviewed by deepseek-v4-flash on August 5, 2026.