{"id":"7dfb1173-7d7e-469e-89bf-8f6b4b0780a4","arxiv_id":"2506.02416","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"Atomic corrections, above all finite-nuclear-size effects on the wave functions, change the bound-muon decay electron spectrum near its endpoint by about 2.5 to 5 percent in carbon, aluminum, and silicon.","lead":"This paper computes the electron energy spectrum from bound-muon decay in carbon, aluminum, and silicon, including finite nuclear size, nuclear deformation, electron screening, vacuum polarization, and nuclear recoil corrections. The result is a refined Standard Model background prediction for upcoming muon-to-electron conversion experiments, where even a few percent shift in the spectrum matters.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Internal inconsistency: Fig. 7 reports a positive total correction for carbon (~5%), but Fig. 1 reports an FNS-only correction of about −44%; these cannot both hold unless the other effects sum to roughly +49%, which the paper's own figures show they do not.","rationale":"The reader identified the kinematic-recoil approximation as the weakest assumption. That approximation is indeed questionable away from the endpoint, but it is applied consistently to both the reference and corrected spectra, so its effect on the relative corrections is likely modest. A more decisive problem is the internal inconsistency between the FNS-only correction and the total correction. The paper's own Fig. 1 shows a large negative FNS correction for carbon and silicon, while Fig. 7 shows a positive total correction of only a few percent. Simple multiplicative combination of the individually reported corrections cannot bridge the gap: the other effects (Ue, SCR, MS) are at the few-percent level, so the total should remain negative at the tens-of-percent level. This discrepancy directly affects the paper's central quantitative claims and undermines confidence in the numerical results. It is also an internal consistency check that the authors could have performed; its absence suggests either a plotting error, a mislabeled axis, a sign error, or a problem with the definition of the total correction. Because the issue is concrete and central, the paper should not be accepted or conditionally accepted without resolution. I therefore recommend UNVERDICTED until the authors clarify or correct the relationship between Fig. 1 and Fig. 7, and ideally release the numerical data or code so that the check can be reproduced.","tokens_in":21001,"tokens_out":33455,"duration_ms":290872,"concrete_test":"For 12C at Ee = 105 MeV, recompute the total correction from the individual corrections multiplicatively: δtot = (1+δFNS)(1+δUe)(1+δSCR)(1+δMS) − 1, using the values from Figs. 1, 4, 6, and the mass-shift binding-energy shift of about 0.95 keV evaluated at the recoil-shifted endpoint. If the result is approximately −0.4, then Fig. 7 is inconsistent; if it is approximately +0.05, then Fig. 1 or the definition of δFNS must be revised. The definitive check is to rerun the full numerical calculation for 12C at 105 MeV with all potentials included and compare the resulting spectrum N_all(E) to the point-nucleus baseline N_pt(E): if N_all/N_pt > 1, Fig. 1 is wrong; if N_all/N_pt < 1, Fig. 7 is wrong.","verdict_should_be":"UNVERDICTED","load_bearing_attack":"The central quantitative claim is that the combined atomic corrections near the endpoint are about 2.5% (Al), 2.8% (Si), and up to 5% (C). Yet Fig. 1 states that the total FNS correction for carbon is approximately −44% close to the endpoint (and −68% for silicon). Since Fig. 7 defines the total correction as the relative deviation of the spectrum with all effects (FNS, Ue, WK, ND, nuclear-recoil, SCR) from the spectrum with none, the total should be roughly (1+δFNS)(1+δUe)(1+δSCR)(1+δMS)−1. Using the paper's own values, δFNS ≈ −0.44, δUe ≈ +0.02, δSCR ≈ +0.04, and the MS energy shift alone contributes no more than a few percent at the plotted energies, giving δtot ≈ −0.40, not +0.05. The small cross terms cannot flip a −40% correction to +5%. Thus Figs. 1 and 7 appear mutually inconsistent. If the FNS correction is indeed −44%, the positive total correction in Fig. 7 and the abstract must be wrong; if the total is truly +5%, then the FNS-only result must be misinterpreted or misplotted. Either way, the paper's headline numbers about the size of atomic corrections are not internally supported.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the electron spectrum from bound-muon decay (decay in orbit) near its endpoint, using Fermi effective theory and a central-field relativistic framework. Two formally equivalent expressions for the spectrum are derived, one based on a two-particle spherical-wave treatment and one based on an effective one-particle current, and the equivalence is reduced to an integral identity in Appendix G. Numerical results are presented for 12C, 27Al, and 28Si, with finite-nuclear-size (FNS), nuclear-deformation (ND), Uehling and Wichmann-Kroll vacuum-polarization, electron-screening (SCR), and nuclear-recoil corrections incorporated into the Dirac equation for the muon and the final electron. The paper reports that the FNS correction alone can be about -44% (C) or -68% (Si) near the endpoint, but that the total correction including all atomic effects is about +2.5% (Al), +2.8% (Si), and up to +5% (C).","tokens_in":21284,"tokens_out":17727,"duration_ms":189870,"significance":"If the numerical results are correct, the paper would establish that finite-nuclear-size effects must be treated self-consistently in the Dirac equation for both the bound muon and the outgoing electron in decay-in-orbit background predictions for upcoming muon-to-electron conversion experiments, and it would provide a detailed two-formula derivation of the spectrum. The work uses independent published inputs (nuclear radii, deformation parameters, masses, potential models), does not fit any parameter to the target spectrum, and benchmarks part of the calculation against prior aluminum results, including the Uehling and screening corrections. These are genuine strengths. However, the central quantitative conclusion is currently undermined by an internal inconsistency between the FNS-only corrections and the claimed total corrections, which must be resolved before the results can be relied upon.","major_comments":[{"comment":"The total FNS correction shown in Fig. 1 (solid curves, defined by Eq. (15)) is about -44% for 12C and -68% for 28Si near the endpoint, while Fig. 7 reports total corrections that include FNS of about +2.5% (Al), +2.8% (Si), and up to +5% (C). Under the stated definitions, these two figures cannot both be correct: the other displayed corrections (Ue about +2.2%, SCR at most a few percent, ND up to about 0.4%, WK negligible) cannot cancel a -44% to -68% FNS term. The paper must either specify a different reference spectrum for Fig. 7 or correct the numbers; as written, the headline total-correction values in the abstract and conclusion are not supported by Fig. 1.","section":"Section III, Figs. 1 and 7"},{"comment":"The kinematic nuclear-recoil correction is introduced by replacing Ee with Ee - Ee^2/(2Mn) in Eq. (11) and Eq. (13), and the text states that this approximation is valid only near the spectrum endpoint. The same replacement is nevertheless applied in all plotted results over the full range from 100 MeV to the endpoint, with no separate uncertainty estimate. Since Ee^2/(2Mn) is about 0.5 MeV for carbon at 100 MeV, the approximation error away from the endpoint could be large and energy dependent; the authors should either restrict the plotted range or quantify the uncertainty introduced by this substitution.","section":"Section III, kinematic recoil paragraph"},{"comment":"The combined 'total correction' plotted in Fig. 7 is not defined by an explicit equation; Eq. (15) defines a relative deviation for a single correction. The reference spectrum for the combined case must be stated precisely (for example, point-nucleus with or without kinematic recoil, and with or without the mass-shift recoil correction). The apparent inconsistency between Figs. 1 and 7 may originate in an ambiguous baseline, and specifying the baseline and the composition of the 'nuclear recoil' term is necessary for the results to be reproducible.","section":"Section III, Eq. (15) and Fig. 7"}],"minor_comments":[{"comment":"The phrase 'unbound sates' should read 'unbound states'.","section":"Section II, Eq. (5)"},{"comment":"The caption lists 27Al among the isotopes, but the figure legend shows only carbon and silicon; either add the aluminum curve or correct the caption.","section":"Fig. 6 caption"},{"comment":"The phrase 'a total total energy correction' contains a duplicated word and should read 'a total energy correction'.","section":"Section III, paragraph after Fig. 7"},{"comment":"The validity of the key integral identity is stated to be 'confirmed numerically', but no precision, convergence criterion, or quadrature parameters are given; adding these details would strengthen the claimed equivalence of Eqs. (9) and (11).","section":"Appendix G, Eq. (G6)"},{"comment":"The text does not explicitly state whether the electron-screening potential used for the unbound electron is the same as that used for the muon, or whether the Z-1 approximation is applied to both particles; clarifying this would help reproducibility.","section":"Section III, screening potential paragraph"}],"recommendation":"major_revision","confidential_remarks":"In my view the inconsistency between Figs. 1 and 7 is the central issue. If the authors cannot resolve it after checking their definitions and recomputing the combined corrections, the paper should not be accepted in its current form. The formal derivation in Section II and the appendices is a useful contribution, and the self-consistent FNS treatment may survive as an important finding, but the quantitative conclusions need to be recomputed and presented with an unambiguous baseline."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Main thing: the paper has a load-bearing internal inconsistency. Fig. 1 shows the total FNS correction alone is about -44% for carbon near the endpoint and -68% for silicon. Fig. 7 and the abstract claim the combined correction is positive, +2.5% to +5%, in the same energy range. That cannot be right unless the other corrections sum to roughly +49% for C and +70% for Si. The paper's own figures show they do not: Ue gives about +2.2%, SCR is sub-percent at plotted energies, MS is sub-percent, ND is 0.4% at most. The kinematic recoil is consistently included in the baseline and cancels in the relative correction, so it cannot rescue the sign. This is not a misreading; either Fig. 1 or Fig. 7/abstract is wrong, and the paper does not address the tension.\n\nWhat is genuinely good: the two independent derivations of the electron spectrum (the 2p and 1p approaches) are detailed and appear correct, and the definite-integral relation in Appendix G is a real piece of new formalism. The computed binding-energy corrections for aluminum agree with earlier literature, giving some confidence in the numerical machinery. The physical point that finite nuclear size must be treated self-consistently in the Dirac equation for both muon and electron, even at Z=6, is plausible and important for the CLFV background calculations.\n\nOther soft spots are minor by comparison: spectral corrections carry no propagated uncertainties, no code or numerical tables are released, and the kinematic recoil replacement Ee -> Ee - Ee^2/(2Mn) is applied across the whole plotted range although the paper states it is valid only near the endpoint. Those are addressable.\n\nWho should read this: people working on muon-to-electron conversion backgrounds will want to know about the FNS self-consistency point, but they should not trust the total correction numbers until the contradiction is resolved. The paper deserves a serious referee—the derivations are substantial—but the referee must force the authors to reconcile the sign and magnitude of the headline results. I would not cite it until that is fixed.","headline":"Careful derivation and a plausible FNS story, but the headline total corrections (+2.5% to +5%) are hard to square with their own FNS-only numbers (-44% for C, -68% for Si); one of those two results is likely wrong.","tokens_in":746,"tokens_out":1703,"would_cite":false,"duration_ms":152331,"reading_group":"maybe","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Finite-nuclear-size effects must be included self-consistently in the Dirac equation for both the bound muon and the outgoing electron; for carbon the correction to the electron spectrum exceeds -40% near the endpoint.","keywords":["bound-muon decay","electron endpoint spectrum","finite-nuclear-size effect","nuclear deformation","vacuum polarization","electron screening","muon-to-electron conversion","charged-lepton flavor violation"],"falsifier":"A high-statistics measurement of the electron spectrum from muon decay in orbit on a carbon target within about 1 MeV of the endpoint could settle the claim, because the self-consistent finite-nuclear-size treatment predicts a pronounced suppression relative to the point-nucleus spectrum, and the observed shape near the endpoint would discriminate between the two. Alternatively, a computation that replaces the $E_e \\to E_e - E_e^2/(2M_n)$ substitution with a full relativistic treatment of the final-state electron kinematics would test the load-bearing approximation away from the endpoint.","tokens_in":20801,"feed_emoji":"⚛️","tokens_out":16160,"duration_ms":141504,"temperature":0.7,"pith_summary":"This paper calculates the electron spectrum from bound-muon decay (muon decay in orbit) near its endpoint for carbon, aluminum, and silicon, the target nuclei of upcoming searches for charged-lepton-flavor-violating muon-to-electron conversion. It argues that finite-nuclear-size, nuclear-deformation, vacuum-polarization, electron-screening, and nuclear-recoil corrections must be inserted as local potentials directly into the Dirac equation, so that they change not only the muon binding energy but also the wave functions of the bound muon and the outgoing electron. The central result is that finite-nuclear-size effects dominate near the endpoint: for carbon-12 the correction exceeds -40%, so treating the nucleus as point-like is inadequate even at low $Z$. Including all effects, the total atomic correction near the endpoint is about 2.5% for aluminum, 2.8% for silicon, and up to 5% for carbon. These numbers set the Standard Model background that next-generation muon-to-electron conversion experiments must subtract.","feed_headline":"Finite nuclear size shifts muon-decay electron spectrum by 40%","feed_subtitle":"Atomic corrections reach 2.5 to 5 percent near the endpoint, where muon-to-electron conversion searches look.","key_machinery":"The load-bearing object is the radial Dirac equation for the bound muon and the unbound electron in the same spherical potential, with each atomic effect added as a local potential: the Fermi nuclear-charge distribution for finite nuclear size, an angle-averaged deformed-Fermi potential for nuclear deformation, Uehling and Wichmann-Kroll potentials for vacuum polarization, an X$\\alpha$-type screening potential for electron screening, and a mass-shift operator for the muon recoil. The electron spectrum is evaluated from two equivalent Fermi-theory expressions, a two-particle multipole expansion and an effective one-particle current, whose equivalence generates a definite-integral relation among spherical Bessel functions. The diagnostic that carries the argument is the split of each correction into an energy part, a wave-function part, and the total; this split reveals which effects must be treated self-consistently and which enter only through the endpoint energy.","core_discovery":"The paper's central claim is that finite-nuclear-size effects on the bound-muon-decay spectrum cannot be captured by a binding-energy shift alone: the finite size of the nucleus must be put into the Dirac equation for both the initial muon and the final electron. With the Fermi nuclear-charge distribution in the potential, the wave-function part of the FNS correction dominates near the endpoint, giving roughly -44% for carbon and -68% for silicon, and the paper concludes that even low-$Z$ nuclei such as $^{12}\\mathrm{C}$ require the fully self-consistent treatment. The same analysis shows that nuclear deformation is significant for silicon but not for carbon, that the Uehling vacuum-polarization correction is a few percent, that the Wichmann-Kroll correction is negligible, and that electron screening acts almost entirely through the endpoint energy. As a by-product, two equivalent formulas for the spectrum for an arbitrary bound muon state are derived, and their equality yields a definite-integral relation among products of spherical Bessel functions. The reported total atomic correction near the endpoint is about 2.5% for aluminum, 2.8% for silicon, and up to 5% for carbon.","pith_inferences":["For heavier nuclei used in other muon-to-electron conversion searches, such as titanium or gold, the finite-nuclear-size correction grows with $Z$, so the self-consistent treatment advocated here would likely modify the endpoint spectrum even more than in the light-nucleus cases shown.","The equivalence of the two expressions for the spectrum is a general statement about four-fermion contact interactions in a central field, so the derived spherical-Bessel integral relation could transfer to other bound-decay calculations such as bound-beta decay.","The conclusion that screening enters only through the energy shift depends on the $Z-1$ approximation for the electron configuration; a different treatment of the atomic environment around the muon could shift the endpoint and deserves its own uncertainty estimate.","If the kinematic recoil substitution were replaced by a full QED recoil treatment, the reported 2.5-5% total corrections could shift by an amount comparable to the mass-shift uncertainty, which future experiments would need as a theory error bar."],"forward_implications":["Background spectra for upcoming muon-to-electron conversion searches in carbon, aluminum, and silicon must be computed with finite-nuclear-size potentials in the Dirac equation for both the muon and the electron; point-nucleus results, even for $Z=6$, are not adequate.","For carbon and silicon, the finite-nuclear-size correction is carried mainly by the wave-function modification, so treatments that only shift the muon binding energy will miss the dominant effect.","Nuclear-deformation corrections are appreciable only for silicon (about 0.4-0.5% near the endpoint) and negligible for carbon, so the correction has to be evaluated isotope by isotope.","Electron screening can be accounted for through the muon binding-energy shift alone, because its wave-function contribution is below 0.01% for the nuclei studied.","The total atomic correction near the endpoint, roughly 2.5% for aluminum, 2.8% for silicon, and up to 5% for carbon, must be folded into the estimated background of next-generation charged-lepton-flavor-violation searches."],"supporting_citations":[{"why":"It is the previous spectrum calculation that assumed a point-like nucleus for oxygen-16, which the present paper argues is inadequate even for low-$Z$ nuclei.","marker":"[24]"},{"why":"It is the earlier aluminum study whose kinematic recoil substitution for the final electron is adopted, and its aluminum binding-energy and screening values serve as comparison.","marker":"[25]"},{"why":"It is the earlier aluminum calculation of Uehling corrections to the spectrum, used to validate the present Uehling results.","marker":"[27]"},{"why":"It supplies the numerical routine for solving the radial Dirac equation on a grid, the core computational tool for both muon and electron wave functions.","marker":"[29]"},{"why":"It provides the Fermi nuclear-charge-distribution parameters, including root-mean-square radii, used to build the finite-nuclear-size potential.","marker":"[30]"},{"why":"It gives the method for constructing the nuclear-deformation potential by averaging a modified Fermi potential over angular coordinates.","marker":"[31]"},{"why":"It generates the local Uehling and Wichmann-Kroll vacuum-polarization potentials used in the Dirac equation.","marker":"[33,34]"},{"why":"It is the source of the estimated uncertainty of the non-relativistic mass-shift recoil correction relative to a full QED treatment.","marker":"[37]"},{"why":"It is the nuclear-mass compilation used for the recoil kinematics of the final-state electron.","marker":"[38]"}],"fun_headline_variants":["Bound-muon decay spectrum shifts up to 5% from atomic effects","Atomic corrections to bound-muon decay reach 5% near endpoint","Finite nuclear size dominates atomic correction in bound-muon decay","Bound-muon decay in atoms: atomic effects shift electron spectrum up to 5%"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the nuclear-recoil correction to the outgoing electron can be modeled by the substitution $E_e \\to E_e - E_e^2/(2M_n)$ in the spectrum, which the paper states is valid only near the endpoint where the electron is effectively massless, yet the substitution is applied across the entire plotted range from 100 MeV to the endpoint without a separate uncertainty assigned to it.","fun_headline_variants_meta":{"raw":{"variants":["Bound-muon decay spectrum shifts up to 5% from atomic effects","Atomic corrections to bound-muon decay reach 5% near endpoint","Finite nuclear size dominates atomic correction in bound-muon decay","Bound-muon decay in atoms: atomic effects shift electron spectrum up to 5%"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001016,"raw_usage":{"total_tokens":4267,"prompt_tokens":903,"completion_tokens":3364,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":519,"completion_tokens_details":{"reasoning_tokens":3281}},"tokens_in":519,"tokens_out":3364,"duration_ms":28441,"temperature":1.0,"reasoning_tokens":3281,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T11:25:08.203471+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A high-statistics measurement of the electron spectrum from muon decay in orbit on a carbon target within about 1 MeV of the endpoint could settle the claim, because the self-consistent finite-nuclear-size treatment predicts a pronounced suppression relative to the point-nucleus spectrum, and the observed shape near the endpoint would discriminate between the two. Alternatively, a computation that replaces the $E_e \\to E_e - E_e^2/(2M_n)$ substitution with a full relativistic treatment of the final-state electron kinematics would test the load-bearing approximation away from the endpoint.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It is the previous spectrum calculation that assumed a point-like nucleus for oxygen-16, which the present paper argues is inadequate even for low-$Z$ nuclei."},{"cited_title":"Tiomno and J","cited_arxiv_id":null,"evidence_quote":"It is the earlier aluminum study whose kinematic recoil substitution for the final electron is adopted, and its aluminum binding-energy and screening values serve as comparison."},{"cited_title":"Tenaglia, Il Nuovo Cimento 13, 284 (1959)","cited_arxiv_id":null,"evidence_quote":"It is the earlier aluminum calculation of Uehling corrections to the spectrum, used to validate the present Uehling results."},{"cited_title":"¨Uberall, Physical Review 119, 365 (1960)","cited_arxiv_id":null,"evidence_quote":"It supplies the numerical routine for solving the radial Dirac equation on a grid, the core computational tool for both muon and electron wave functions."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It provides the Fermi nuclear-charge-distribution parameters, including root-mean-square radii, used to build the finite-nuclear-size potential."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It gives the method for constructing the nuclear-deformation potential by averaging a modified Fermi potential over angular coordinates."},{"cited_title":"Watanabe, M","cited_arxiv_id":null,"evidence_quote":"It is the source of the estimated uncertainty of the non-relativistic mass-shift recoil correction relative to a full QED treatment."},{"cited_title":"Watanabe, K","cited_arxiv_id":null,"evidence_quote":"It is the nuclear-mass compilation used for the recoil kinematics of the final-state electron."}],"review_version":1}