{"id":"d2327061-d92c-477f-ac59-a97e363579b4","arxiv_id":"1908.09562","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"For 0.14 to 6 GeV heavy neutrinos, accounting for the parent meson's velocity weakens projected LNV meson decay sensitivities by one to two orders of magnitude at NA62 and SHiP.","lead":"This paper calculates how often rare lepton number violating meson decays could be spotted in six running or planned experiments, and how those rates change when the decaying meson is moving. The results tell experimenters how sensitive their detectors really are to a hypothetical heavy neutrino that could explain neutrino masses.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Finite-detector probability omits the target-to-detector baseline; for MATHUSLA and SHiP the survival factor can change the computed velocity-effect reach.","rationale":"The reader's verdict is CONDITIONAL, and my concern supports keeping it CONDITIONAL rather than moving it. The paper's strongest claim is not simply that HNL searches in LNV meson decays are sensitive, but that parent-meson velocity changes the reach by one to two orders of magnitude and that the meson-at-rest approximation is unreliable. Both Eqs. (12)-(13) and every figure inherit the decay-probability factor from Eq. (8). That factor is a one-dimensional probability with the detector starting at the production point. The paper itself notes MATHUSLA's 100 m displacement (Sec. 7.4), and SHiP is a beam dump with a decay volume after the absorber; in both cases there is a finite baseline over which the N must survive before it can decay in the fiducial volume. Omitting exp(-L_in/L_N) makes P_N artificially remain of order unity when L_N is comparable to or smaller than L_in, exactly the regime sampled on the large-mixing side of the reach curves and near kinematic endpoints. Thus the absolute normalization of the projected reaches and the size of the claimed velocity effect are not yet fixed by the calculation; they depend on an idealized geometry. This is not an attack on the direction of the effect, which is physically reasonable, but on the quantitative claim being load-bearing. The reader's concern about zero background and 100% efficiency is valid, but the missing baseline is more directly tied to the paper's own finite-detector formalism and was not flagged as a caveat in Sec. 8. A single rerun with the survival factor included would settle it, so the appropriate verdict remains CONDITIONAL pending that check.","tokens_in":29580,"tokens_out":26288,"duration_ms":294586,"concrete_test":"Recompute the MATHUSLA and SHiP reach curves (Figs. 7-10) with P_N = exp(-L_in/L_N)[1 - exp(-L_D/L_N)], using L_in = 100 m for MATHUSLA and the SHiP target-to-decay-volume distance (~60 m) with L_D as the decay-volume length. Compare the resulting |V_eN|^2 and |V_muN|^2 lines at M_N = 2, 3, and 5 GeV, and also recompute the at-rest versus in-flight ratio in Fig. 6. If the shifted curves and the ratio change by less than ~30%, the baseline omission is a minor normalization; if they change by an order of magnitude, the quoted reaches and the claimed velocity-effect factor require revision.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The finite-detector probability used throughout Eqs. (8), (12), (13), and all figures is P_N = 1 - exp(-L_D/L_N), which treats the detector as a one-dimensional segment beginning at the N production point. This is not the geometry of the experiments that drive the headline reach. Sec. 7.4 states that MATHUSLA is displaced ~100 m from the IP, and SHiP's decay volume sits downstream of the hadron absorber and muon shield; in both cases the N must survive a baseline L_in before it can decay in the fiducial volume. The correct probability is exp(-L_in/L_N)[1 - exp(-L_D/L_N)]. Omitting exp(-L_in/L_N) makes P_N saturate to 1 when L_N is short, whereas the real detector sees almost no signal because the N decays before arrival. This changes event counts by orders of magnitude on the large-mixing side of every reach curve and can shift the inferred sensitivity lines, especially near kinematic thresholds where L_N is short. Because the paper's central claim that parent-meson velocity changes the reach by one to two orders of magnitude is obtained by comparing two evaluations of this same incomplete P_N, the quantitative statement is not yet demonstrated for the actual experimental geometries. The Sec. 8 caveat mentions detection and reconstruction efficiencies, but not this geometric baseline.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":29820,"tokens_out":9603,"duration_ms":101001,"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":[{"comment":"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.","section":"§4, Eqs. (6)–(10); §6"},{"comment":"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.","section":"§6; §8 caveat; Abstract"},{"comment":"","section":null}],"minor_comments":[{"comment":"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.","section":"§5"},{"comment":"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.","section":"Fig. 5"},{"comment":"The sentence 'Even with mixing angle equal to 1' should read 'mixing squared equal to 1', since the plotted quantity is |VℓN|2.","section":"Fig. 3"},{"comment":"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.","section":"§7.1 and §7.5"},{"comment":"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.","section":"Eq. (13)"}],"recommendation":"major_revision","confidential_remarks":"The paper is within the scope of a phenomenological journal and the central physics idea is sensible. The main technical problem is the missing production-to-detector baseline in the decay probability, which is fixable by rerunning the calculation with the experimental geometries. The authors should also clarify the democratic-mixing convention and either soften the abstract/conclusion claims or carry the idealization caveat into the abstract. I do not see grounds for rejection, but the quantitative sensitivity curves need revision before publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: the velocity effect is real and the paper is right to insist on it. For D, D_s and K decays at SHiP and NA62, treating the parent meson as at rest changes the projected mixing reach by one to two orders of magnitude. That claim survives reading the paper. What does not survive is the absolute normalization: the numbers quoted in the abstract—SHiP reaching |V_eN|^2 ~ 10^-9, MATHUSLA and LHCb ~ 10^-7—come from an idealized calculation.\n\nWhat is new: a channel-by-channel menu of LNV meson decays M->ll pi for six experiments and six mixing combinations, using the boosted decay probability. The amplitude and narrow-width formulas are standard and correctly applied. The width calculation is transparent, and the paper does state the zero-background and 100-percent-efficiency assumption in Sec. 8.\n\nSoft spots, in rough order of severity. First, the detector probability in Eqs. (8) and (13) is 1 - exp(-L_D/L_N). For MATHUSLA and SHiP, the N must first survive from the IP or target to the decay volume; the correct factor is exp(-L_in/L_N)[1 - exp(-L_D/L_N)]. Omitting the survival factor makes the large-mixing side of every reach curve optimistic, and it is not mentioned in the Sec. 8 caveat list. Second, zero background and 100 percent efficiency are assumed for all channels; the caveat acknowledges this, but the curves are still plotted as sharp sensitivities. Third, the total width is normalized with |V_eN|^2 = |V_muN|^2 = |V_tauN|^2, so the quoted bounds on individual mixings are convention-dependent. Fourth, fixed meson momenta and a one-dimensional detector are fine for a first pass but not for experiment-level claims. No code or data files are provided; that is minor here, since the formulas are standard.\n\nOn the stress-test note: it is right about the missing baseline. I would push back only on the implication that the central comparison is invalidated. The at-rest versus in-flight difference is genuine for the idealized geometry, and in some cases the baseline makes that difference larger, not smaller. But the absolute SHiP and MATHUSLA lines should not be used without a proper geometry and efficiency treatment.\n\nWho this is for: HNL phenomenologists and experimental colleagues planning displaced-vertex searches. It is a useful first-pass menu and a legitimate extension of the Gorbunov-Shaposhnikov and Bondarenko et al. program. I would send it to peer review, asking for a revised version that includes the baseline survival factor and presents efficiencies and backgrounds as bands rather than sharp lines.","headline":"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.","tokens_in":30356,"tokens_out":4500,"would_cite":true,"duration_ms":49357,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["14.60.St","13.20.Fc","14.60.Pq"],"model":"deepseek-v4-flash","headline":"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 $10^{-7}$.","keywords":["lepton number violation","heavy Majorana neutrino","right-handed neutrino","active-sterile mixing","long-lived particle searches","SHiP","MATHUSLA","meson decays"],"falsifier":"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.","tokens_in":29349,"feed_emoji":"🔭","tokens_out":14248,"duration_ms":113039,"temperature":0.7,"pith_summary":"This paper argues that projected searches for a heavy Majorana neutrino in lepton-number-violating meson decays—decays that produce two same-sign charged leptons and change lepton number by two—must account for the parent meson's motion and the finite detector size, and that the common meson-at-rest approximation misstates the reach by one to two orders of magnitude. The authors model decays such as $D_s^-\\to e^-e^-\\pi^+$ and $K^-\\to e^-e^-\\pi^+$ mediated by an on-shell right-handed neutrino $N$ of mass $0.14$-$6$ GeV, and compute the probability that $N$ decays inside SHiP, MATHUSLA, NA62, LHCb, Belle II, and FCC-ee. With the velocity effect included, the future beam-dump experiment SHiP can probe electron-sterile mixing down to $|V_{eN}|^2\\sim 10^{-9}$, while MATHUSLA gives the best reach around $|V_{eN}|^2\\sim 10^{-7}$ for $N$ masses between 2 and 5 GeV. The point of the calculation is that the same decay chain yields looser bounds when the parent meson is moving, so experiment projections and existing exclusion plots should be read with the boost, not the rest frame, in mind.","feed_headline":"Meson velocity weakens neutrino-search reach by 10-100x","feed_subtitle":"Treating parent mesons as at rest overstates the sensitivity; boosted parents lower it by one to two orders.","key_machinery":"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.","core_discovery":"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}