{"id":"e74051b0-a26e-417f-a250-efec5c29cdc4","arxiv_id":"2411.16231","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"The decay of structured neutrons is an incoherent sum of plane-wave decays, and the proton angular distribution inherits the neutron state's geometry, including a forward dip for vortex neutrons and discrete rotational symmetries for spin-orbit states.","lead":"This paper calculates how the radioactive decay of a neutron changes when the neutron is prepared in a twisted or structured quantum state rather than a simple plane wave. It finds that the angular distribution of the decay's proton is a sensitive fingerprint of the neutron's spatial structure, which could be used to characterize such states in future experiments.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The spin-orbit asymmetry magnitudes are computed with a pure V–A Hamiltonian (g_A=g_V=1); the real weak current alters spin correlations substantially, so the quantitative 'tool to extract features' claim is not yet supported.","rationale":"The reader's formal weakest assumption was the practical unavailability of fast structured neutron beams, but the reader's rationale also noted that the quantitative spin-orbit asymmetries rely on a simplified weak interaction. My stress-test focuses on that latter point, which is more fundamental because it undermines the numerical predictions even in the energy regime where structured beams currently exist. The kinematic effects for unpolarized Bessel and LG neutrons—the critical-angle threshold at about 750 eV, the forward dip, and the ring at the cone opening angle—are phase-space effects and are robust to the weak-interaction model. The spin-orbit azimuthal asymmetry, however, is a spin-correlation observable; its size in Figs. 8–11 is controlled by the spin-dependent part of the decay amplitude. In the real Standard Model the hadronic axial coupling is g_A ≈ -1.2756, not 1, and recoil-order corrections are known to change spin observables by large factors. Thus the paper's central quantitative claim for the spin-orbit sector is not firmly established. This does not invalidate the paper's qualitative conclusion—the angular distribution should inherit the discrete symmetry of the spin density—but it does mean the specific asymmetry values and the claimed observability at 10 meV are conditional on a full Standard Model recomputation, or on explicitly softening the claim to a qualitative symmetry statement. I therefore agree with the reader's CONDITIONAL verdict and do not propose changing it; the added condition is that the full weak-interaction amplitude must be used before the numerical tool claim is taken at face value. The unresolved factor-of-2 discrepancy with Ref. [15] is also worth resolving, but it is a normalization issue that would cancel in the azimuthal ratios central to the spin-orbit claim, so I do not treat it as the single most load-bearing concern.","tokens_in":14591,"tokens_out":16600,"duration_ms":175166,"concrete_test":"Recompute the spin-orbit proton azimuthal distributions of Figs. 8–11 using the full low-energy charged-current amplitude with g_A = -1.2756 and including recoil-order weak-magnetism and induced-pseudoscalar corrections, while keeping all other parameters and kinematic cuts identical. Compare the maximum relative azimuthal variation at each neutron energy (10 meV, 0.5 keV, 4.5 keV); if any of these amplitudes changes by more than about 30% or changes sign, the paper's quantitative observability claims must be revised, even though the C_N pattern may survive.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The quantitative claims about spin-orbit neutron decay—the C_N azimuthal modulation of the proton distribution and its magnitude in Figs. 8–11, including the statement that the effect 'can possibly be observed even at currently available energies (~10 meV)'—are based on the simplified Fermi interaction in Eq. (10), where the vector and axial-vector couplings are taken equal (g_A=g_V=1). The spin-dependent substitution (24), p_n^mu -> p_n^mu - m_n s^mu, and its cross-term analogue (47) are exact only for this pure left-handed contact interaction. The real low-energy charged current has g_A ≈ -1.2756 and contains recoil-order weak-magnetism and induced-pseudoscalar terms, which materially change spin-correlation observables: for example, the beta-asymmetry parameter is roughly -0.27 experimentally, whereas g_A=1 gives -1. Since the spin-orbit azimuthal asymmetry is a spin-correlation observable built from interference of opposite-spin amplitudes, its computed magnitude—especially the 5e-4 variation at 10 meV—is not trustworthy until recomputed with the full Standard Model amplitude. The discrete rotational pattern is likely robust, because it follows from the symmetries of the spin density, but the quantitative 'tool to extract features' claim rests on numbers that the simplified Hamiltonian cannot reliably provide.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies the beta decay of a free neutron prepared in non-plane-wave states: Bessel (vortex) states, Laguerre-Gaussian wave packets, and spin-orbit superpositions of LG modes. The authors derive the plane-wave decay rate in the laboratory frame, then show that for Bessel and general wave packets the decay rate reduces to an incoherent azimuthal average of plane-wave rates (Eqs. (29), (32)), while for spin-orbit states an additional coherent cross-term survives (Eq. (43)). The main predictions are: (i) the proton spectral-angular distribution is highly sensitive to the neutron wave-packet structure, developing a forward dip and a ring at the vortex cone angle for neutron kinetic energies above about 750 eV; and (ii) the proton azimuthal distribution inherits the discrete rotational symmetry of the neutron spin density, with a magnitude that the authors suggest may be observable at 10 meV. The paper concludes that proton SAD can be used as a tool to extract features of structured neutron states.","tokens_in":14862,"tokens_out":16882,"duration_ms":191759,"significance":"The conceptual framework is sound: the incoherent averaging over transverse momenta is standard in the twisted-particle literature, and the derivation from first principles avoids fitted parameters. The kinematic ring/dip effect for Bessel neutrons is a concrete, falsifiable prediction that is independent of the details of the weak Hamiltonian. The spin-orbit symmetry-reflection result (C_N pattern in the proton azimuthal distribution) is a novel extension of the muon-decay analysis of Ref. [18] to neutron decay and is likely robust because it follows from the spin-density symmetries. The main limitation is that all numerical magnitudes are computed with a simplified V-A contact interaction (g_A=g_V=1, no recoil-order terms), and the paper does not quantify the resulting uncertainty; this affects the quantitative claim of observability at 10 meV, though not the qualitative symmetry and kinematic predictions.","major_comments":[{"comment":"The quantitative spin-orbit predictions are derived from the pure left-handed Fermi contact interaction of Eq. (10) with g_A=g_V=1 and no recoil-order terms. The spin projection rule (24) and the cross-term substitution (47) are exact only for this interaction. The real charged current has g_A/g_V ≈ -1.2756 as well as weak magnetism and induced pseudoscalar form factors, which substantially change spin-correlation observables; for example, the beta-asymmetry parameter is about -0.118 experimentally, whereas g_A=1 gives -1. Since the azimuthal asymmetry in Figs. 8-11 is a spin-correlation observable built from interference of opposite-spin amplitudes, the stated magnitudes (including the 5×10^-4 variation at 10 meV and the claim that the effect 'can possibly be observed even at currently available energies') are not quantitatively supported. The discrete rotational pattern likely survives because it follows from the symmetry of the spin density, but the paper should recompute the magnitudes with the full Standard Model amplitude or clearly label the numbers as illustrative for the simplified interaction.","section":"§III.A, Eq. (10); §IV, Figs. 8–11 and Conclusion"},{"comment":"Equation (4) as printed places the denominator inside the square root, making the expression dimensionally inconsistent. For m=m_p and \\tilde m=m_e, the printed formula gives β_cr ≈ 1.68, not the quoted β_cr ≈ 0.00126. The correct form is \\beta_{cr} = \\sqrt{[(m_n+\\tilde m)^2-m^2][(m_n-\\tilde m)^2-m^2]}/(m^2+m_n^2-\\tilde m^2), i.e., the square root should apply only to the numerator product. The numerical values used later (750 eV) are consistent with the corrected formula, so this is likely a typographical error, but it must be fixed because the equation as printed cannot be used.","section":"§II, Eq. (4)"}],"minor_comments":[{"comment":"The phrase 'where 𝓁 is the is the azimuthal quantum number' contains a duplicated 'is the'; it should read 'where 𝓁 is the azimuthal quantum number.'","section":"Eq. (26) text"},{"comment":"The statement that differences between structured and plane-wave decays are 'solely due to kinematics' is an overstatement: the spin-orbit cross-term of Eq. (43) is a quantum interference effect controlled by the relative phase b and the spin-density symmetry, not a purely kinematic effect.","section":"Sec. V, Conclusion"},{"comment":"The matching condition σ_p = (2/3) p_z tan θ_n chosen to compare LG and Bessel states is ad hoc; the sensitivity of the qualitative comparisons to this choice is not discussed and should be at least briefly stated.","section":"Figs. 3, 5, 6"},{"comment":"The sentence calling Eq. (10) 'the standard model calculation' is misleading; the actual amplitude is the Fermi contact interaction with g_V=g_A=1, which differs from the Standard Model low-energy charged current at the level of the axial coupling and recoil-order terms. This should be reworded for clarity.","section":"Sec. III.A"}],"recommendation":"major_revision","confidential_remarks":"The paper is already published in Physical Review C (DOI 10.1103/PhysRevC.111.024619), and the present review is for a different venue. The main technical concerns are the simplified weak Hamiltonian and the typo in Eq. (4). The novelty relative to Ref. [15] lies mainly in the spin-orbit states; the Bessel/LG results are incremental. The paper would benefit from a recomputation with the real g_A/g_V ratio and recoil-order corrections before the quantitative observability claim is stated."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear Colleague,\n\nQuick take: this is a solid, clearly written theoretical paper on beta decay of structured neutrons. The genuinely new results are the LG wave-packet treatment, the spin-orbit decay calculation, and the proton critical-angle effect (forward dip and ring at the cone angle for Bessel neutrons above ~750 eV). The paper also corrects the energy-range claim in the earlier Bessel calculation [15]. The formalism is standard twisted-particle incoherent averaging, and the derivation in the appendices is self-contained and easy to follow. The argument that the proton SAD is far more sensitive than the electron SAD is convincing and should survive any interaction-model changes.\n\nThe main soft spot is quantitative. The spin-orbit azimuthal asymmetries are computed with a pure V-A contact interaction (g_A = g_V = 1, no form factors). The paper says this explicitly, but the numbers in Figs. 8-11—especially the 5e-4 modulation at 10 meV—are not robust to the real weak current. With g_A ≈ 1.275, the spin-dependent interference terms shift by tens of percent or more, and weak-magnetism/recoil terms, though small, can enter at a similar order for some of the discussed observables. The discrete rotational pattern (C_N symmetry) is a symmetry consequence of the initial spin density and should remain, but the magnitude is not a reliable experimental prediction as it stands. If the paper wants the 'tool to extract features' claim to be taken literally, it needs a computation with the full Standard-Model amplitude or at least a quantitative bound on the g_A sensitivity.\n\nSecond, there is an unresolved factor-of-2 disagreement with the previous Bessel-neutron decay paper [15]. The authors note that their rate is twice smaller but do not identify which calculation is correct. That matters for anyone using these rates as a reference.\n\nOn feasibility: the proton ring/dip effects require structured neutron beams at keV energies that do not exist yet; the paper concedes generation methods are unexplored. The 10 meV spin-orbit effect is in reach of current beams, but its magnitude is exactly the one most sensitive to the simplified interaction. So the experimental case is mixed: a clean kinematic prediction waiting for fast beams, and a symmetry demonstration whose calibration is still open.\n\nWho this is for: people working on neutron optics, twisted-particle decay, and beta-decay theory. I'd bring it to a reading group as a good example of wave-packet decay calculations. I would cite the kinematic critical-angle result and the LG treatment, but not the spin-orbit magnitudes without a caveat. It definitely deserved a serious referee (and got one, PRC). For engagement: push the authors on the g_A dependence and the factor-of-2; the qualitative effects are likely right.","headline":"The proton ring/dip prediction is solid and useful; the spin-orbit asymmetry magnitudes need a full weak-interaction treatment before being used quantitatively.","tokens_in":15384,"tokens_out":5015,"would_cite":true,"duration_ms":114780,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Proton angular patterns from structured neutron decay reveal the shape of the decaying wave packet.","keywords":["neutron beta decay","vortex neutrons","orbital angular momentum","spin-orbit coupled states","Bessel beams","Laguerre-Gaussian wave packets","spectral-angular distribution","weak decay of structured states"],"falsifier":"Look for the predicted proton patterns: for a 75 keV Bessel neutron with cone opening angle $\\theta_n\\approx0.1$ rad, the proton angular distribution should show a forward minimum and a peak at $\\theta_p\\approx0.1$ rad; if it matches the plane-wave forward cone instead, the critical-angle mechanism fails. Similarly, protons from a 0.5 keV spin-orbit neutron with $\\Delta\\ell=-1$ and $b=0$ should show a two-peaked $C_2$ azimuthal pattern, and its absence would rule out the symmetry-inheritance claim.","tokens_in":1775,"feed_emoji":"🌀","tokens_out":3530,"duration_ms":92794,"temperature":0.7,"pith_summary":"The paper argues that a free neutron prepared as a non-plane-wave \"structured\" state—a Bessel vortex beam, a Laguerre-Gaussian wave packet, or a spin-orbit superposition—decays in a way that leaves a readable imprint on the outgoing proton's energy and angle. Its central claim is that the proton's spectral-angular distribution is highly sensitive to the structure of the neutron wave packet, and can therefore serve as a tool to extract features of structured neutron states. Concretely, Bessel neutrons above about 750 eV kinetic energy produce a forward dip and a ring-shaped peak in the proton angular distribution, while spin-orbit neutron states imprint their discrete rotational spin symmetry onto the proton azimuthal distribution. Electron distributions are comparatively insensitive at feasible energies. The paper presents this as a diagnostic for structured neutron beams and as evidence that weak decays reflect the spatial symmetries of decaying quantum states.","feed_headline":"Proton sprays expose vortex neutron structure","feed_subtitle":"Neutron beta decay imprints the shape of the decaying wave packet onto proton angles, offering a new beam diagnostic.","key_machinery":"The load-bearing mechanism is the incoherent azimuthal average of plane-wave decay widths together with the lab-frame kinematic critical angle. The Lorentz boost from the neutron rest frame introduces a maximal proton deflection angle for neutron velocities above $\\beta_{\\rm cr}\\approx0.00126$ (kinetic energy about 750 eV), and each plane-wave component of a Bessel state sees a different effective direction, so the averaged proton distribution develops a dip at zero angle and a ring at the vortex cone angle. Analytically, the twisted decay width reduces to $d\\Gamma_{\\rm tw}=\\int (d\\phi_n/2\\pi)\\,d\\Gamma_{\\rm PW}$, and the spin-orbit cross term is evaluated through the substitution $p_n^\\mu\\to m_n(n_1^\\mu\\pm i n_2^\\mu)$, which carries the spin-density symmetry into the final proton distribution.","core_discovery":"The authors compute the $\\beta$-decay rate for an arbitrary initial neutron wave packet and find that different plane-wave components of the packet contribute incoherently: the differential rate is an average of plane-wave rates weighted by the packet's momentum density. For an unpolarized Bessel neutron this reduces the decay width to an azimuthal average over the vortex cone, making the result independent of the orbital angular momentum quantum number. The proton distribution nonetheless differs sharply from the plane-wave case once the neutron's kinetic energy exceeds about 750 eV, where a kinematic critical angle exists: each plane-wave component can only emit protons within a cone, and averaging over the Bessel ring produces a forward minimum and a peak at the cone opening angle. For Laguerre-Gaussian packets the OAM enters through the transverse probability density, broadening the proton spectrum, while for spin-orbit states the diagonal and cross terms transfer the spin-density field's discrete rotational symmetry, of order $|\\Delta\\ell-1|$ and rotated by the relative phase $b$, into the proton azimuthal distribution.","pith_inferences":["Editorial extension: the same kinematic critical-angle argument should apply to any three-body decay of a fast vortex particle when the detected daughter is nearly as heavy as the parent, making the proton-versus-electron contrast a direct test of that geometry.","Editorial extension: the rotation of the proton azimuthal pattern with the relative phase $b$ could be used as a weak-decay polarimetric measurement of the prepared spin-orbit state.","Editorial extension: the paper does not estimate event rates, so realizing the 10 meV spin-orbit signal would require large decay counts or a brighter structured neutron source; this is a practical question left open.","Editorial extension: if fast structured neutron generation matures, the forward-ring pattern could serve as a beam diagnostic for vortex neutron flux and opening angle without interferometric detection."],"forward_implications":["Proton spectral-angular distributions can act as a diagnostic of non-plane-wave neutron states, revealing the vortex cone opening angle and the transverse width of the packet.","For Bessel neutrons with kinetic energy above about 750 eV, the proton angular distribution shows a forward dip and a peak at the cone opening angle, while electron distributions lack this feature at experimentally feasible energies.","For spin-orbit neutron states, the proton azimuthal pattern inherits the discrete rotational symmetry of the neutron spin density, with order $|\\Delta\\ell-1|$ and a phase shift set by the relative phase $b$ of the two modes.","Bessel-state decays are independent of the neutron OAM value, whereas Laguerre-Gaussian decays depend on OAM through the probability density, so the two kinds of structured states are distinguishable by their proton spectra.","The spin-orbit azimuthal asymmetry is only about $5\\times10^{-4}$ at 10 meV but grows to roughly 0.2 at 0.5 keV, so the effect could be seen with existing slow structured neutron beams only with high statistics."],"supporting_citations":[{"why":"Earlier calculation of twisted neutron decay that this work extends and uses as the plane-wave versus vortex comparison baseline.","marker":"[15]"},{"why":"Introduces the spin-orbit neutron state formalism as superpositions of Laguerre-Gaussian modes with opposite spins.","marker":"[7]"},{"why":"Reports experimental generation of spin-orbit neutron states, motivating the 10 meV examples in the paper.","marker":"[8]"},{"why":"Shows that vortex muon decay rates reduce to azimuthal averages of plane-wave decay widths, a pattern this paper follows.","marker":"[17]"},{"why":"Demonstrates that emitted particle angular distributions inherit the discrete rotational symmetry of the initial polarization state.","marker":"[18]"},{"why":"Supplies the regularization and normalization of twisted states used to reduce the Bessel decay rate to the azimuthal average.","marker":"[28]"},{"why":"Provides the phase-space integration technique used to derive the plane-wave differential decay rate with massive final particles.","marker":"[25]"},{"why":"Documents the capabilities of existing fast neutron sources, justifying the paper's exploration of neutron energies up to 300 MeV.","marker":"[24]"}],"fun_headline_variants":["Proton angles map the neutron wave packet shape","Twisted neutrons stamp their shape on proton spray","Neutron vortex shape seen in beta decay protons","Proton angles expose neutron's wave packet geometry","Beta decay protons trace the neutron's twist"],"cache_read_input_tokens":17536,"weakest_assumption_plain":"The diagnostic value rests on having neutron beams that are both structured and fast enough: the ring effect needs kinetic energies above about 750 eV, while at currently available 1–10 meV the spin-orbit azimuthal modulation is only about $5\\times10^{-4}$, and the paper states that methods for generating fast structured neutrons are not yet explored.","fun_headline_variants_meta":{"raw":{"variants":["Proton angles map the neutron wave packet shape","Twisted neutrons stamp their shape on proton spray","Neutron vortex shape seen in beta decay protons","Proton angles expose neutron's wave packet geometry","Beta decay protons trace the neutron's twist"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000194,"raw_usage":{"total_tokens":1334,"prompt_tokens":903,"completion_tokens":431,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":519,"completion_tokens_details":{"reasoning_tokens":360}},"tokens_in":519,"tokens_out":431,"duration_ms":4852,"temperature":1.0,"reasoning_tokens":360,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T13:21:12.814795+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Look for the predicted proton patterns: for a 75 keV Bessel neutron with cone opening angle $\\theta_n\\approx0.1$ rad, the proton angular distribution should show a forward minimum and a peak at $\\theta_p\\approx0.1$ rad; if it matches the plane-wave forward cone instead, the critical-angle mechanism fails. Similarly, protons from a 0.5 keV spin-orbit neutron with $\\Delta\\ell=-1$ and $b=0$ should show a two-peaked $C_2$ azimuthal pattern, and its absence would rule out the symmetry-inheritance claim.","supporting_citations":[{"cited_title":"Afanasev, V","cited_arxiv_id":null,"evidence_quote":"Earlier calculation of twisted neutron decay that this work extends and uses as the plane-wave versus vortex comparison baseline."},{"cited_title":"Sarenac, J","cited_arxiv_id":null,"evidence_quote":"Introduces the spin-orbit neutron state formalism as superpositions of Laguerre-Gaussian modes with opposite spins."},{"cited_title":"Sarenac, C","cited_arxiv_id":null,"evidence_quote":"Reports experimental generation of spin-orbit neutron states, motivating the 10 meV examples in the paper."},{"cited_title":"Nsofini, D","cited_arxiv_id":null,"evidence_quote":"Shows that vortex muon decay rates reduce to azimuthal averages of plane-wave decay widths, a pattern this paper follows."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Demonstrates that emitted particle angular distributions inherit the discrete rotational symmetry of the initial polarization state."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the regularization and normalization of twisted states used to reduce the Bessel decay rate to the azimuthal average."},{"cited_title":"The coefficients are calculated as follows: I αβ (q2gαβ − 2qα qβ ) = 4Bq4 = ∫ [( p¯νe pi ) q2 − 2 ( qp ¯νe ) (qpi ) ]d3 pi ωi d3 p¯νe ω¯νe","cited_arxiv_id":null,"evidence_quote":"Provides the phase-space integration technique used to derive the plane-wave differential decay rate with massive final particles."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Documents the capabilities of existing fast neutron sources, justifying the paper's exploration of neutron energies up to 300 MeV."}],"review_version":1}