REVIEW 3 major objections 5 minor 2 cited by
Sensitivity of Lepton Number Violating Meson Decays in Different Experiments
T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash
Pith's one-line read Including the parent meson's motion in lepton-number-violating meson-decay searches changes the projected reach by one to two orders of magnitude, with SHiP probing $|V_{eN}|^2\sim10^{-9}$ and MATHUSLA $10^{-7}$.
desk verdict Right that parent-meson boost matters for LNV meson-decay sensitivities, but the projected reaches are idealized and the decay-probability formula omits the production-to-detector baseline for SHiP and MATHUSLA. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing object is the detector-decay probability $P_N=1-\exp(-L_D/L_N)$, with $L_N=p_N/(M_N\Gamma_N)$, together with the boost transformation of the neutrino momentum: in the parent-meson rest frame $p_N^*$ is fixed, but for a meson of momentum $p_{M_1}$ the neutrino energy ranges over $E_N\in[\gamma E_N^*\mp p_N^*\sqrt{\gamma^2-1}]$ with a flat distribution. The paper feeds this $E_N$ range into its master event-counting formula, replacing the rest-frame $P_N$ by an integral over $P_N'=1-\exp(-M_N\Gamma_N L_D/\sqrt{E_N^2-M_N^2})$. This single change—boosting the intermediate neutrino before asking whether it decays in a detector of finite length—is what shifts the projected mixing reach by one to two orders of magnitude. The total width $\Gamma_N=\sum_{\ell}a_\ell(M_N)|V_{\ell N}|^2$ is computed at tree level over pseudoscalar, vector, and three-body channels so the narrow-width factorization stays valid.
What would settle it
A concrete check is to take the measured production-momentum spectrum of $D_s$ mesons at a 400 GeV proton beam dump and recompute the SHiP reach with a full three-dimensional detector geometry and a realistic background count. If the exact boost-averaged decay probability differs from the single-momentum $p_{D_s}=58$ GeV result by less than an order of magnitude, or if even one background event appears in the $D_s\to ee\pi$ signal region, the projected $|V_{eN}|^2\sim 10^{-9}$ reach will not survive.
Extended reading notes
Core claim
The central claim is that a right-handed Majorana neutrino $N$ in the mass window $0.14\ \text{GeV}\le M_N\le 6\ \text{GeV}$ can be resonantly produced in the three-body LNV decays $M_1^-\to\ell_1^-\ell_2^-\pi^+$ for $M_1=B,B_c,D,D_s,K$, and that the sensitivity of current and proposed experiments to the active-sterile mixings $|V_{\ell N}|^2$ and $|V_{\ell_1 N}V_{\ell_2 N}|$ depends strongly on treating these mesons as moving. The decay rate factorizes under the narrow-width approximation into $\Gamma(M_1\to\ell_1 N)\times \mathrm{Br}(N\to\ell_2\pi^+)$, and the observable signal requires $N$ to decay inside the detector. Because the decay length $L_N=p_N/(M_N\Gamma_N)$ grows with the neutrino momentum, a boosted parent meson produces a harder $N$ spectrum and a smaller decay probability $P_N=1-e^{-L_D/L_N}$; integrating the flat energy distribution of $N$ in the lab frame loosens the bounds by one to two orders of magnitude, most sharply for $D$, $D_s$ at SHiP and $K$ at NA62. With that correction, SHiP's $D_s\to ee\pi$ channel reaches $|V_{eN}|^2\sim 10^{-9}$, MATHUSLA's $B\to ee\pi$ reaches $|V_{eN}|^2\sim 10^{-7}$ for $2\ \text{GeV}<M_N<5\ \text{GeV}$, and LHCb's $B_c$ channels give the tightest tau-mixing constraints in the 5-6 GeV range.
Load-bearing premise
The projected numbers assume zero background events, perfect detection and reconstruction efficiency, a one-dimensional detector length with no angular acceptance, and equal couplings of $N$ to electron, muon, and tau; real detectors that fail any of these will have weaker sensitivities.
Editorial extensions
If this is right
- Rest-frame projections are unreliable for the forward, boosted experiments: NA62, LHCb, and SHiP each see looser mixing bounds when parent-meson velocity is included, by roughly one order for $B$ and $D_s$ and two for $K$.
- SHiP's $D_s^-\to e^-e^-\pi^+$ and $D_s^-\to\mu^-\mu^-\pi^+$ channels can probe $|V_{eN}|^2,|V_{\mu N}|^2\sim 10^{-9}$ for $0.14\ \text{GeV}<M_N<1.9\ \text{GeV}$.
- For $2\ \text{GeV}<M_N<5\ \text{GeV}$, MATHUSLA's $B\to\ell\ell\pi$ modes give the strongest reach, $|V_{eN}|^2,|V_{\mu N}|^2\sim 10^{-7}$, while LHCb's $B_c\to\ell\ell\pi$ modes dominate above 5 GeV.
- Tau-flavor mixings $|V_{\tau N}|^2$ and the products $|V_{eN}V_{\tau N}|$, $|V_{\mu N}V_{\tau N}|$ can be probed at the $10^{-7}$ level by $B,B_c\to\tau\tau\pi,\ e\tau\pi,\ \mu\tau\pi$ modes at MATHUSLA and LHCb, in mass ranges largely unconstrained by other searches.
- For $\ell_1\ne\ell_2$, the same-sign final state is both lepton-number and lepton-flavor violating, and the long $N$ lifetime allows the two orderings $M_1\to\ell_1\ell_2\pi$ and $M_1\to\ell_2\ell_1\pi$ to be distinguished experimentally.
Reading between the lines
- The same boost-averaging correction should apply to any long-lived particle produced in meson decays, not just Majorana neutrinos: a fixed boost or rest-frame approximation will tend to overestimate the visible-decay probability whenever the decay length is comparable to the detector size.
- The paper's fixed parent momenta (58 GeV for SHiP, 75 GeV for NA62, 100 GeV for LHCb) are proxies; folding in actual production-momentum spectra could widen or narrow the spread, so the rest-versus-in-flight comparison should be repeated per momentum bin before quoting a single combined reach.
- If the heavy neutrino is a quasi-Dirac pair rather than a pure Majorana state, the $\Delta L=2$ amplitude is suppressed by the mass splitting, so these searches are really measuring the Majorana component; the quoted limits are a maximal-Majorana benchmark rather than a general sterile-neutrino bound.
- A testable extension: apply the same formalism to heavy neutrinos produced from $W/Z$ decays at MATHUSLA and FCC-ee, where the parent momentum distribution differs, to see whether the velocity correction reshapes those projected reaches as much as it does for meson decays.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. This paper studies lepton-number-violating three-body meson decays M1− → ℓ1−ℓ2−π+ mediated by an on-shell right-handed Majorana neutrino N in the mass range 0.14–6 GeV. It computes event rates for NA62, LHCb, Belle II, SHiP, MATHUSLA, and FCC-ee, including the effect of the parent meson boost and a finite detector length, and derives projected sensitivities on |VeN|2, |VμN|2, |VτN|2, and the products |VeNVμN|, |VeNVτN|, |VμNVτN|. The central quantitative claim is that including the parent meson velocity changes the inferred mixing reach by one to two orders of magnitude, with SHiP probing |VeN|2 ∼ 10−9 at low mass and MATHUSLA/LHCb reaching ∼10−7 near 5 GeV.
Significance. The paper is a useful and clearly structured phenomenological survey of LNV meson-decay probes, with explicit partial-width formulas in the appendix and a broad comparison against existing and projected constraints. The observation that parent-meson boost can significantly modify the effective decay length is physically sensible and relevant to how simplified projections are made. The authors are also transparent about several idealizations: the zero-background 3.09-event benchmark, 100% detection and reconstruction efficiency, and the democratic-mixing normalization in the total width are all stated explicitly. If the quantitative results survive a more realistic detector treatment, the paper would provide valuable guidance for the heavy-neutrino search programs at the facilities considered. However, the current sensitivity curves are conditional on a simplified one-dimensional, no-baseline detector model that is not adequate for all of the experiments that drive the headline results.
major comments (3)
- [§4, Eqs. (6)–(10); §6] The total decay width ΓN is evaluated with the ad hoc democratic normalization |VeN|2 = |VμN|2 = |VτN|2. Since ΓN appears both in the lifetime and in the branching ratio Γ(N→ℓ2π)/ΓN, the derived limit on a single mixing element such as |VeN|2 from Ds→eeπ is not a bound on that element alone, but a bound in a specific three-flavor-equal scenario. The paper states this assumption explicitly, but it should be justified or its impact quantified, because hierarchical mixings are typical in realistic seesaw realizations and could shift the quoted reaches substantially.
- [§6; §8 caveat; Abstract] The projected limits assume zero background and 100% detection and reconstruction efficiency, with the average upper limit set to 3.09 events at 95% C.L. The caveat paragraph in §8 correctly acknowledges the idealized detector, but the abstract and conclusions quote the resulting numbers as the experimental 'sensitivity reach' without this qualification. The headline statements should be explicitly conditional on these assumptions, and for experiments such as NA62 and LHCb the zero-background assumption should at least be motivated by a background estimate. This is a presentation issue as much as a physics issue, but it directly affects how the numbers in the abstract are read.
minor comments (5)
- [§5] The text contains several typographical slips, such as 'compare' for 'compared' in the discussion of meson decay in flight; these should be corrected in a revised version.
- [Fig. 5] The vertical axis of Fig. 5 is labeled with the symbol ϰ, while the text defines the variable as x = L_D/L_N; the figure should use the same symbol as the text or explicitly define ϰ.
- [Fig. 3] The sentence 'Even with mixing angle equal to 1' should read 'mixing squared equal to 1', since the plotted quantity is |VℓN|2.
- [§7.1 and §7.5] The Bc meson yield at LHCb and some Belle II yields are based on private communications, as acknowledged in footnotes 74 and 84; it would improve reproducibility to give the resulting yields with stated uncertainties or to cite public sources where available.
- [Eq. (13)] The flat distribution in E_N follows from isotropy in the parent-meson rest frame; the text should state this assumption explicitly and note that angular correlations in the three-body decay are neglected.
Circularity Check
No circularity found: the sensitivity reaches are computed from standard model amplitudes, an explicit decay-width calculation, and external experimental event-number inputs.
full rationale
The derivation chain is self-contained: (i) the LNV amplitude in Eq. (4) follows from the standard charged-current Lagrangian; (ii) the total width in Eq. (7) is computed in the Appendix from explicit tree-level partial widths; (iii) event counts are built from Eqs. (12)-(13) using the decay probability in Eq. (8); (iv) limits are obtained by equating the event count to the external Feldman-Cousins benchmark of 3.09 events at 95% C.L. In each step, the mixing V_lN is the unknown parameter being solved for, not a fitted input renamed as a prediction. The democratic choice |V_eN|^2=|V_muN|^2=|V_tauN|^2 in the total width is an explicitly stated assumption, and it approximately cancels in the branching-ratio factor in the short-decay-length regime where the SHiP and NA62 reach is quoted. The paper's many self-citations are contextual and do not carry the central velocity-effect comparison. The Sec. 8 caveat about idealized 100% detection/reconstruction efficiency and the separate geometric-baseline concern about omitting exp(-L_in/L_N) are quantitative modeling limitations, not circular reductions, because the same incomplete probability is used as an input in both the at-rest and in-flight comparisons rather than being derived from the quantity the paper claims to predict.
Assumptions & free parameters
free parameters (2)
- Democratic mixing scale in Gamma_N =
|VeN|^2=|VmuN|^2=|VtauN|^2=1
- Background event count =
0
assumptions (8)
- domain assumption One light right-handed Majorana neutrino in the 0.14 to 6 GeV mass range mediates the LNV decays, with two heavier right-handed neutrinos assumed irrelevant.
- standard math Narrow-width approximation for the on-shell heavy neutrino.
- domain assumption Light neutrino exchange in the LNV meson decays is negligible.
- ad hoc to paper Democratic mixing normalization in Gamma_N when deriving limits.
- ad hoc to paper Zero background and 100 percent detection and reconstruction efficiency.
- ad hoc to paper Detector geometry is a single length with no angular acceptance.
- domain assumption Fixed parent meson momenta for SHiP, NA62, and FCC-ee.
- domain assumption Meson production yields are taken from experimental projections and private communications without error bars.
invented entities (1)
-
Right-handed Majorana neutrino N
independent evidence
Cite this review
Pith. "Pith review of Sensitivity of Lepton Number Violating Meson Decays in Different Experiments." pith.science (2026). https://pith.science/paper/7CNLU3DH
@misc{pith2026190809562,
author = {Pith},
title = {Pith review of: Sensitivity of Lepton Number Violating Meson Decays in Different Experiments},
year = {2026},
howpublished = {\url{https://pith.science/paper/7CNLU3DH}},
note = {Machine review of arXiv:1908.09562}
}
abstract
We study the discovery prospect of different three body lepton number violating~(LNV) meson decays $M_{1}^{-}\to\ell_{1}^{-}\ell_{2}^{-}M_{2}^{+}$ in the framework of right handed~(RH) neutrino extended Standard Model~(SM). We consider a number of ongoing experiments, such as, NA62 and LHCb at CERN, Belle II at SuperKEK, as well as at the proposed future experiments, SHiP, MATHUSLA and FCC-ee. The RH Majorana neutrino $N$ mediating these meson decays provides a resonant enhancement of the rates, if the mass of $N$ lies in the range $(100\, \text{MeV}-6\, \text{GeV})$. We consider the effect of parent mesons velocity, as well as, the effect of finite detector size. Using the expected upper limits on the number of events for the LNV decay modes, $M_{1}^{-} \to\ell_1^{-}\ell_2^{-}\pi^{+}$~($M_{1}=B, B_c,D, D_{s}\,\text{and}\,K$), we analyze the sensitivity reach of the mixing angles $|V_{e N}|^{2}$, $|V_{\mu N}|^{2}$, $|V_{\tau N}|^{2}$, $|V_{e N}V_{\mu N}|$, $|V_{e N}V_{\tau N}|$ and $|V_{\mu N}V_{\tau N}|$ as a function of heavy neutrino mass $M_{N}$. We show that, inclusion of parent meson velocity can account to a large difference for active-sterile mixing, specially for $D$, $D_s$ meson decay at SHiP and $K$ meson decay at NA62. Taking into account the velocity of the $D_s$ meson, the future beam dump experiment SHiP can probe $|V_{eN}|^2 \sim 10^{-9}$. For RH neutrino mass in between 2 - 5 GeV, MATHUSLA can provide best sensitivity reach of active-sterile mixings.
Figures
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Forward citations
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Reference graph
Works this paper leans on
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[1]
The solar and atmo- spheric mass splittings areO(10−5) andO(10−3) eV2, while the three mixings are θ12∼ 33◦, θ23∼ 45◦ and θ13∼ 9◦
INTRODUCTION The discovery of neutrino oscillations [1] in a series of oscillation experiments have con- firmed that neutrinos have non-zero masses and non-zero mixings. The solar and atmo- spheric mass splittings areO(10−5) andO(10−3) eV2, while the three mixings are θ12∼ 33◦, θ23∼ 45◦ and θ13∼ 9◦. These observations indicate, at least two of the three SM...
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THE MODEL We extend the SM to include additional RH neutrinos N. The heavy neutrinos can generate light neutrino masses through seesaw. For simplicity, we consider only one RH neutrino and carry out our analysis. The mixing of N with the active neutrinos are given by the following expression, ν𝓁 = 3∑ m=1 U𝓁mνm +V𝓁NNc m′, (1) where νm and Nm′ are the mass ...
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The Feynman diagrams for these decays are shown in Figs
PROCESS The RH neutrino N, if a Majorana state can mediate the LNV process M− 1 →𝓁− 1𝓁− 2M + 2 . The Feynman diagrams for these decays are shown in Figs. 1 and 2. The diagram in Fig. 2 will give a very small contribution, as this is not a resonance production diagram. Note 4 q1 q2 W − q3 q4 𝓁− 1 𝓁− 2 W + Ni M − 1 M + 2 q1 q2 W − q3 q4 𝓁− 2 𝓁− 1 W + Ni M −...
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For the mass range 0 .140 GeV≤MN≤ 6 GeV, RH neutrino can be produced as an intermediate on mass shell state in the LNV meson decays being considered here
TOT AL DECA Y WIDTH OF N Although the RH neutrino N is a SM singlet, due to mixing with active neutrinos it can decay via charged and neutral current interactions. For the mass range 0 .140 GeV≤MN≤ 6 GeV, RH neutrino can be produced as an intermediate on mass shell state in the LNV meson decays being considered here. We consider only tree level diagrams i...
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The RH neutrino produced in meson decays M1→ 𝓁1N, propagates and decays after traveling some distance from the production point
P ARENT MESON VELOCITY AND FINITE DETECTOR SIZE EFFECT For the mass range 0 .140 GeV≤ MN ≤ 6 GeV, the RH neutrino produced in these LNV meson decays are on shell. The RH neutrino produced in meson decays M1→ 𝓁1N, propagates and decays after traveling some distance from the production point. This is the decay length LN of the RH neutrino N and it depends o...
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SIGNAL EVENTS The sensitivity reach for the LNV decay modes in a particular experiment depends on the number of the parent mesons M1’s produced (NM− 1 ), their momentum ( ⃗ pM1) and the branching ratio for these mesons to the LNV modes. Assuming the parent meson M1 decays at rest, the expected number of signal events is [68]: Nevent = 2NM− 1 Br ( M− 1 →𝓁−...
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LHCb The LHCb detector is a forward spectrometer at the Large Hadron Collider (LHC) at CERN
INPUT FOR DIFFERENT EXPERIMENTS 7.1. LHCb The LHCb detector is a forward spectrometer at the Large Hadron Collider (LHC) at CERN. A search for heavy Majorana neutrinos inB→µµπ decay mode had been performed by the LHCb collaboration using 7 TeV data [70] and bound on the mixing angle |VµN|2 is provided in the mass range 0 .25 GeV≤ MN≤ 5 GeV1. The cross-sec...
2024
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6, we show how the velocity of the parent mesons affect the sensitivity reach of the mixing angles
RESUL TS In Fig. 6, we show how the velocity of the parent mesons affect the sensitivity reach of the mixing angles. We consider a number of ongoing and future experiments, such as, FCC-ee, SHiP to explore B→ eeπ, SHiP for Ds→ eeπ, NA62 for K→ eeπ, and LHCb, SHiP for Bc→ ττπ meson decays. To derive the bounds/sensitivity on the mixing angle as a function o...
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The B mesons produced at FCC-ee will have an energy distribution peaked at EB+ = MZ 2
The number of charged B mesons from Z decays can be estimated as, NB+ =NZ× Br ( Z→b¯b ) ×fu, where NZ ∼ 1013, Br ( Z→b¯b ) = 0.1512 [86], fu = 0.410 [87] is the fraction of B+ from ¯b quark in Z decay. The B mesons produced at FCC-ee will have an energy distribution peaked at ...
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