New Avenues for $|\Delta B|$ = 2 Processes Beyond Neutron-Antineutron Oscillations

Alexey A. Petrov; Arnau Bas i Beneito; Svjetlana Fajfer

REVIEW 3 major objections 4 minor 84 references

New Avenues for $|\Delta B|$ = 2 Processes Beyond Neutron-Antineutron Oscillations

T0 review · 3 major / 4 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read Dinucleon decay, not hyperon oscillations, sets the tightest bound on strangeness-changing baryon-number violation.

desk verdict Useful, carefully scoped EFT/chiral paper: the pp→K+K+ bound on δmΛ is the strongest current probe and the ordering is likely robust, but the headline number is an order-of-magnitude estimate with unquantified hadronic uncertainties. read the letter →

arxiv 2511.05657 v2 pith:UFPXBNKZ submitted 2025-11-07 hep-ph

Arnau Bas i Beneito , Svjetlana Fajfer , Alexey A. Petrov This is my paper · ORCID

classification hep-ph

keywords baryonnumberviolationΛ–Λ̄oscillationsneutron–antineutrondinucleondecaysix-quarkoperatorsSMEFTchiralperturbationtheorydimension-9

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper extends baryon-number-violating (ΔB=2) physics beyond neutron–antineutron oscillations to the strange baryon Λ, showing that Λ–Λ̄ mixing can be generated at tree level by six-quark operators of the form (uds)^2 that are independent of the (udd)^2 operators behind n–n̄ oscillations. Using the Standard Model Effective Field Theory and scalar-mediated UV completions, the authors derive indirect bounds on the effective Λ mass-mixing parameter δm_Λ from existing limits on n–n̄ oscillations and from the dinucleon decay p p → K+ K+. They find that the dinucleon channel dominates by far, constraining δm_Λ below about 10^-32 GeV — roughly fourteen orders of magnitude stronger than the direct bound from J/ψ→Λ Λ̄ oscillations. The paper thereby establishes pp→K+K+ searches as the most powerful current probe of strangeness-violating ΔB=2 transitions, sensitive to new-physics scales up to ~300 TeV, and argues that Λ–Λ̄ oscillation experiments would need roughly four orders of magnitude improvement to become competitive.

What carries the argument

The central object is the effective mass mixing δm_Λ, the off-diagonal element in the two-state Hamiltonian for Λ and its antiparticle, generated by six-quark operators (uds)^2 with Wilson coefficients suppressed by the fifth power of a new-physics scale Λ_BNV. The argument is carried by chiral effective theory, which connects δm_Λ to observables: a tree-level diagram converts δm_Λ into n–n̄ mixing via weak vertices, pion and kaon loops give subleading contributions, and a tree-level t-channel diagram produces the amplitude for p p→K+ K+. That amplitude, combined with a nuclear-density estimate of the intranuclear rate, is what turns the dinucleon lifetime limit into the sharp bound on δm_Λ.

What would settle it

A lattice QCD computation of the Λ→Λ̄ six-quark matrix element that found it to be many orders of magnitude smaller than the neutron value would invalidate the claim that pp→K+K+ constrains δm_Λ at the 10^-32 GeV level; conversely, an improved bound or a first positive signal in pp→K+K+ would directly test the prediction.

Watch

Extended reading notes

Core claim

The paper's central claim is that strangeness-violating |ΔB|=2 baryon-number violation is currently best probed by the dinucleon decay p p→K+ K+, not by Λ–Λ̄ oscillations. Starting from the Standard Model Effective Field Theory, the authors identify 52 six-quark operators of the (uds)^2 type (up from 14 for (udd)^2), classify their tree-level scalar-mediated UV completions, and identify models in which Λ–Λ̄ mixing is generated at tree level while n–n̄ mixing appears only at two loops. Using chiral effective theory to bridge quark operators and baryon observables, they derive indirect bounds on δm_Λ from the measured n–n̄ oscillation limit and from the water-Cherenkov detector's limit on p p→

Load-bearing premise

The quantitative bound on δm_Λ assumes the six-quark matrix element for the Λ equals the lattice neutron value up to O(1) factors, and that logarithmically divergent one-loop chiral integrals are regulated with a hard cutoff at 5 GeV; if those hadronic estimates are severely wrong, the numerical hierarchy among bounds shifts, though the ordering probably survives.

Editorial extensions

If this is right

If correct, pp→K+K+ searches at water-Cherenkov detectors currently probe |ΔB|=2 at effective scales up to about 300 TeV, far beyond direct collider reach.
Models with (uds)^2 operators can yield Λ–Λ̄ oscillations at tree level while n–n̄ mixing is loop-suppressed, so the two channels probe genuinely different operator directions.
Improving the pp→K+K+ limit by an order of magnitude would push the new-physics scale into the PeV region.
Λ–Λ̄ oscillation experiments would need to improve δm_Λ sensitivity by roughly four orders of magnitude to compete with collider mass bounds, which appears infeasible in the foreseeable future.
The classification of 52 LEFT operators and 20 SMEFT operators provides a working basis for future lattice computations of the hadronic matrix elements.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If a lattice QCD computation of the Λ six-quark matrix element deviates from the neutron value by more than an order of magnitude, the quantitative hierarchy among bounds would shift, though the qualitative ordering would likely survive because the dinucleon bound is so much stronger.
The same chiral machinery could be applied to Ξ− or Ω− hyperon oscillations, or to ΔB=2 processes with charm quarks, extending the operator catalogue to (ucs)^2 or mixed-flavour combinations.
A dedicated re-analysis of existing water-Cherenkov data on pp→K+K+ using modern nuclear matrix elements would be the cheapest experimental test of the paper's central claim.
The two-loop, GIM-like suppression of n–n̄ relative to Λ–Λ̄ in the simplified model suggests a generic way to hide n–n̄ oscillations while leaving Λ–Λ̄ or dinucleon channels observable.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Here's the short version: this is a genuinely useful paper for the BNV subfield. It gives the first systematic EFT/chiral treatment of Λ–Λ̄ oscillations, classifies the (uds)^2 dimension-9 operators (52 LEFT / 20 SMEFT), adds a previously neglected class of trilinear scalar UV completions, and—most importantly—shows that Super-K's pp→K+K+ limit already constrains δmΛ far more strongly than BESIII's direct oscillation search. The qualitative hierarchy is almost certainly right; the fourteen-order gap to BESIII is enormous.

What's new and good: the operator counting, the µX1X2X3 completions, and the simplified model where Λ–Λ̄ is tree-level and n–n̄ is two-loop. The chiral translation of n–n̄, BESIII, and pp→K+K+ limits into common bounds on δmΛ is exactly the kind of apples-to-apples comparison the field needs. The paper is careful about the proton-decay constraints on scalar LQs and about the GIM-like suppression in the two-loop n–n̄ amplitude.

Soft spots, in order of importance. First, the pp→K+K+ bound rests on the assumption that the two-nucleon→two-kaon amplitude is saturated by the δmΛ pole diagram. That is not forced by EFT: the short-distance operator has independent matrix elements between two-nucleon and two-kaon states. The alternative nuclear estimate (Eq. 3.32) makes this explicit by setting C_i⟨KK|O|pp⟩ ≡ δmΛ, and if that matrix element differs from the single-baryon one, the bound shifts by exactly that ratio. It would take a large suppression to matter given the fourteen-order gap, so I would not bet against the ordering, but the number 10^-32 GeV has no uncertainty attached and is an order-of-magnitude estimate by the authors' own admission. Second, the Λ matrix element is just set equal to the neutron lattice value (Eq. 2.13); there is no lattice input for the strange system. Third, the operator basis is asserted via Sym2Int but not actually listed—the 52/20 counts are not checkable from the paper. Fourth, the chiral loops are regulated with a hard cutoff at 5 GeV, which is above the usual BχPT validity range, and Eq. (3.22) quotes a dimensionless coefficient with a GeV unit.

I am not worried about circularity or data-manipulation; the bounds are derived from external limits and standard LECs. The paper would benefit from a referee insisting that the authors (1) present the explicit operator lists or deposit them, (2) attach an honest error bar to the δmΛ bounds, and (3) fix the typo and explain the matching assumption more carefully.

Verdict: send to peer review. It's a solid subfield contribution that will be cited and should be on the radar of anyone working on baryon-number violation or planning dinucleon searches. I'd bring it to a reading group.

Referee Report

3 major / 4 minor

Summary. The paper extends the effective-field-theory treatment of |ΔB|=2 baryon-number violation from neutron–antineutron oscillations to strangeness-violating Λ–Λbar oscillations. It classifies dimension-9 LEFT and SMEFT operators relevant for the (uds)^2 sector, enumerates scalar-mediated UV completions (trilinear, quartic, and fermionic topologies), and derives indirect bounds on the mass-mixing parameter δmΛ from n–nbar searches, from Super-Kamiokande pp→K+K+ searches, and from BESIII. In a simplified model with the scalars Sbar1 and Ω4, the paper finds that n–nbar is induced only at two loops while Λ–Λbar appears at tree level, and that pp→K+K+ gives the strongest lower bound on the effective BNV scale, Λ_BNV ≳ 300 TeV. The central claim is that current Λ–Λbar oscillation searches are far from competitive with dinucleon-decay bounds.

Significance. If the hadronic matrix-element assumptions are controlled, this is a useful and original phenomenological guide. The operator classification goes beyond prior n–nbar studies, with machine-checked enumeration via Sym2Int and a systematic survey of UV completions; the appendices on fermionic and quartic completions are valuable. The simplified model is explicit and falsifiable: it predicts that pp→K+K+ at Super-K, not BESIII, is the best probe of strangeness-violating |ΔB|=2 physics. The paper also usefully identifies the need for lattice calculations of ⟨Λ|O|Λ⟩ and two-meson nuclear matrix elements. I find no circularity: the bounds are derived from external experimental limits, with the simplified-model bounds being legitimate parameter translations. The main weakness is that the headline numerical hierarchy relies on hadronic matrix-element identifications that are assumed rather than derived, and the paper gives no uncertainty budget.

major comments (3)

[Sec. 3.2, Eqs. (3.27)–(3.30)] The headline bound δmΛ ≲ 1.8×10^-32 GeV from pp→K+K+ is not a model-independent constraint. The BχPT amplitude in Eq. (3.27) inserts the single-baryon mass-mixing operator δmΛ into a long-distance pole diagram. In the hadronic EFT, the same six-quark operator also matches onto local NN→KK operators whose coefficients are not fixed by δmΛ. Eq. (3.32) makes the implicit identification explicit: C_i⟨K+K+|O_i|pp⟩ ≡ δmΛ. If the two-hadron matrix element differs from ⟨Λ|O|Λ⟩, the quoted bound shifts by exactly that ratio. Since the paper presents this as the central result and uses it in Sec. 4 and Table 4, please provide an estimate or a conservative range for R = ⟨K+K+|O|pp⟩/⟨Λ|O|Λ⟩, and show how the claimed fourteen-order separation from BESIII changes under plausible variations. The label 'order-of-magnitude estimate' is not sufficient when the numerical separation is the main quantitative
[Sec. 3.2, Eqs. (3.31)–(3.35)] The alternative nuclear estimate is not internally consistent as written. In Eq. (3.32), M_A = δmΛ ρpp(0) with ρpp(0) = κ|ρ_p(0)|^2: with densities in fm^-6 and δmΛ in GeV, this amplitude has dimension mass^7. The phase-space integral is quoted in Eq. (3.34) as 4×10^-7 GeV^-1, whereas the standard three-body phase space for this decay has dimension mass^2, and no fm↔GeV conversion factors are displayed in the rate formula. Consequently Eq. (3.35) cannot be reproduced from Eq. (3.31) by a dimensionally consistent calculation. This is load-bearing because Table 5 presents the nuclear estimate as independent confirmation of the chiral estimate. Please rewrite with explicit dimensions and numerical conversions, or clearly label Eq. (3.35) as a parametric dimensional estimate.
[Secs. 2.2, 3.1 and Tables 4–5] The numerical bounds are quoted without any uncertainty budget. The unknown Λ matrix element is replaced by the lattice neutron value in Eq. (2.13), with only a symbolic O(m_u,d/m_s) correction; the one-loop integrals in Eqs. (3.13) and (3.21) are regulated with an ad hoc 5 GeV cutoff; and the chiral couplings a,b in Eq. (3.8) carry fit errors that are not propagated. Since the paper's conclusion is quantitative—fourteen orders of magnitude—please provide a conservative error range for each bound, or at least show how the bounds in Tables 4 and 5 depend on the key inputs. This pass should also fix the dimension/step inconsistency in Eq. (3.22): the quantity δm_n^K is written as 9.8×10^-16 GeV, but the subsequent bound δm_Λ^K ≲ 1.4×10^-18 GeV requires that this be interpreted as the dimensionless coefficient 9.8×10^-16 times δmΛ.

minor comments (4)

[Eq. (3.29) and Fig. 5] The amplitude in Eq. (3.27) drops the /q+mΛ numerator of the Λ propagator. This may be an O(1) approximation for nonrelativistic kinematics, but it should be stated. The factor of 2 multiplying 2mp for the u-channel contribution is also not explained in the text.
[Sec. 2.2] The Λ operator basis is not displayed; the reader is told that the number of independent operators is 52 (LEFT) and 20 (SMEFT), with the derivation left to a forthcoming publication. A representative subset or an ancillary file would make the counting reproducible and easier to check.
[Figure 9] The figure caption does not identify the colored regions or the axes beyond 'parameter space'. Please add a legend and define the plotted quantity (e.g., the effective scale Λ_BNV versus scalar mass).
[Sec. 3.1] The hard-cutoff dependence of the one-loop bounds is not discussed. Varying Λχ between ~1 and 5 GeV changes the coefficients in Eqs. (3.14) and (3.22), and the resulting δmΛ bounds are close enough to the BESIII limit that the comparison should include this variation.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular derivation: all δmΛ bounds are translations of independent experimental limits; only minor non-load-bearing self-citations.

full rationale

The derivation chain is not circular. The δmΛ limits are obtained by translating independent external limits (ILL/Super-K τ_{n−n̄}, BESIII τ_{Λ−Λ̄}, Super-K τ(pp→K+K+)) through a chiral Lagrangian; δmΛ is never fitted to itself. Eq. (3.10) converts the measured δm_n bound into δmΛ through weak ΔS=1 vertices with external LECs a,b, and Eqs. (3.14)/(3.22) are explicitly cutoff-regulated one-loop estimates that are ancillary. Eq. (3.27) computes pp→K+K+ from a δmΛ insertion and Eq. (3.30) turns the Super-K lifetime into an upper bound on δmΛ; this is an EFT translation, not a tautology, though it rests on the model-dependent saturation assumption that the short-distance NN→KK operator is dominated by the Λ−Λ̄ pole diagram. The alternative nuclear estimate (3.31)–(3.35) sets C_i⟨K+K+|O_i|pp⟩ρ_pp(0) ≡ δmΛ ρ_pp(0); this is an explicitly labelled 'order-of-magnitude estimate' of a missing hadronic/nuclear input, not a self-referential fit, because the input on the right is the independent experimental width. The main acknowledged weakness is Eq. (2.13), where ⟨Λ|O_i|Λ⟩ is equated to the neutron lattice value; this is a missing hadronic computation (also flagged in the text as 'not yet available'), not a circular step. The self-citations [44,71,72] support standard CKM-suppression remarks, an existing SU(5) discussion, and a heavy-neutrino caveat; none carries the central derivation, so the score stays at 1.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The paper introduces no new particles; all exotic scalars and fermions are taken from previous literature (Table 2 and Appendices). The main external inputs are the SMEFT operator basis, chiral low-energy constants, the neutron lattice matrix element used for the Λ, and a set of nuclear modeling approximations. The free parameters listed above are the quantitative handles that set the numerical bounds.

free parameters (5)

Chiral LECs a and b in the |ΔS|=1 Lagrangian = a = 1.68×10^-8 GeV, b = -4.26×10^-8 GeV
Fitted to hyperon nonleptonic decays; enter the tree-level and loop relations between δmn and δmΛ (Eqs 3.8–3.14).
Hard cutoff Λχ for divergent chiral loops = 5 GeV
Regulates logarithmically divergent pion and kaon loop integrals; results shift with the chosen cutoff near the chiral-symmetry-breaking scale.
Λ hadronic matrix element ⟨Λ̄|O|Λ⟩ = 10^-5 GeV^6
Borrowed from lattice neutron matrix elements and assumed equal for the Λ (Eq 2.13); used in all numerical bound conversions.
Nuclear density ρN and kaon momentum |k| = ρN ≈ 0.25 fm^-3, |k| ≈ 0.2 GeV
Inputs to the pp→K+K+ rate estimate (Eqs 3.28–3.29); chosen to represent oxygen-nucleus conditions at order-of-magnitude level.
Short-range correlation κ and point-proton density ρp(0) = κ ≈ 4, ρp(0) ≈ 0.08 fm^-3
Central values used in the alternative three-body nuclear estimate 16O→14C K+K+ (Eqs 3.33–3.35).

assumptions (6)

standard math The dimension-9 SMEFT ΔB=2 operator basis from Refs [46,47] is complete.
The paper relies on this classification to generate the (uds)^2 operators and to discuss UV completions.
domain assumption The Λ–Λ̄ hadronic matrix element equals the neutron lattice value (Eq 2.13).
Explicitly stated; SU(3) breaking is treated as an O(mu,d/ms) correction but no numerical error is assigned.
domain assumption Lowest-order BχPT (Eq 3.7) and the weak |ΔS|=1 Lagrangian (Eq 3.8) dominate the hadronic transitions.
All chiral relations between δmn, δmΛ, and pp→K+K+ use this truncation; higher-order terms and counterterms are not included.
ad hoc to paper Log-divergent loop integrals are regulated with a hard cutoff at 5 GeV.
Introduced by hand in Sec 3.1; the resulting pionic and kaonic bounds depend on this prescription.
domain assumption Intranuclear pp→K+K+ rate is estimated from the free-proton amplitude using an average nucleon density or ρpp(0)=κ|ρp(0)|^2.
Nuclear model connecting Super-K's 16O→14C K+K+ bound to the quark-level operator; no full nuclear calculation is available.
domain assumption In the simplified model, n–n̄ arises only at two loops via the stated CKM/GIM-suppressed diagram (Sec 4.1).
The complementarity claim depends on this suppression being numerically correct; other operators or completions could alter the hierarchy.

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Pith. "Pith review of New Avenues for $|\Delta B|$ = 2 Processes Beyond Neutron-Antineutron Oscillations." pith.science (2026). https://pith.science/paper/UFPXBNKZ

@misc{pith2026251105657,
  author       = {Pith},
  title        = {Pith review of: New Avenues for $|\Delta B|$ = 2 Processes Beyond Neutron-Antineutron Oscillations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UFPXBNKZ}},
  note         = {Machine review of arXiv:2511.05657}
}

abstract

We explore baryon-number-violating ($|\Delta B| = 2$) processes beyond the well-known neutron-antineutron ($n - \bar{n}$) oscillations, focusing on the $\Lambda - \bar \Lambda$ system. The presence of a strange quark in the $\Lambda$ baryon introduces a new set of six-quark operators roughly of the form $(uds)^2$, which are different from the $(udd)^2$ operators responsible for $n - \bar{n}$ oscillations. Using the Standard Model Effective Field Theory (SMEFT), we classify all dimension-9 operators that cause $|\Delta B|=2$ transitions and study their UV completions mediated by exotic scalar fields with trilinear interactions. We demonstrate that in these models, $\Lambda - \bar \Lambda$ oscillations can occur at tree level, with $n - \bar{n}$ mixing potentially appearing at higher loop levels. We employ a chiral effective theory to constrain the effective mass mixing $\delta m_\Lambda$, deriving bounds from current experimental limits on $n - \bar{n}$ oscillations and dinucleon decays such as $p \,p \to K^+ K^+$. These bounds indicate that $\Lambda - \bar{\Lambda}$ oscillations probe a complementary parameter space, sensitive to baryon-number violation at scales up to $10^2-10^3$ TeV. We show that the existing indirect bounds make it challenging to provide a competitive bound on $\delta m_\Lambda$ at BESIII.

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