{"id":"3096526a-7da4-46c4-8a75-8112e144a0d1","arxiv_id":"2505.20587","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"No-core shell model calculations for tritium show radial excitation contributions to the isospin-symmetry-breaking correction are negative and comparable to 10-20% of the diagonal contribution, suggesting shell-model δC2 values are likely overestimated.","lead":"This paper calculates how radial excitations affect the isospin-symmetry-breaking correction in Fermi beta decays, using exact no-core shell model calculations for tritium. It finds the effect is negative and typically 10 to 20 percent of the main correction, and that including it makes the Standard Model fit to superallowed beta decays worse, not better.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The paper's global conclusions rest on an unvalidated mapping of a tritium-specific, truncation-dependent κ to all 15 superallowed emitters, so the matching/extraction procedure that defines κ is the load-bearing assumption.","rationale":"The reader's weakest assumption (nucleus-independent κ) is real, but it is not by itself decisive: all NCSM-derived κ values in Table I are negative, and since δsm_C2 is positive in the Hardy-Towner survey, any negative κ increases Ft_i, which is the direction that worsens the CVC+CKM test. Thus the qualitative sign of the Standard-Model shift is robust to relaxing the single-κ assumption as long as κ remains negative. The deeper problem is that the value of κ itself—including its sign and magnitude—is an output of a one-parameter matching of the NCSM convergence deficit to a nonorthogonal harmonic-oscillator calculation, and that matching has not been validated against an independent radial treatment (e.g., Woods-Saxon) or against a second nucleus. The strong Nmax and ℏωi dependence, especially at the physically preferred frequencies, means the 'typical 10–20%' is not a well-defined observable. This is why I regard the matching/transferability issue, rather than the fixed-κ simplification alone, as the most load-bearing concern. The Table I factor-of-10 inconsistency is secondary but should be corrected. The conclusions are plausible and appropriately hedged, but they remain conditional pending this validation; the reader's CONDITIONAL verdict is appropriate.","tokens_in":14408,"tokens_out":17887,"duration_ms":204419,"concrete_test":"Using the same Woods-Saxon single-particle prescription as Hardy-Towner for each of the 15 superallowed emitters, compute the full radial-overlap correction including both n = n' and n ≠ n' terms; the resulting per-nucleus κ_i = δre_C2/δsm_C2 directly tests whether the tritium-derived range applies. If the κ_i are not systematically negative at the −10% to −20% level, the central generalization fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central quantitative claims depend on identifying the NCSM convergence deficit δC − δC1(Nmax) with the nonorthogonal-basis δC2(Δℏω) and then reading off κ = δre_C2/δsm_C2 at the fitted Δℏω. This identification is a modeling assumption, not a derivation: every finite-Nmax deficit can be reproduced by choosing some Δℏω, and the diagonal-versus-excitation split is then fixed by the harmonic-oscillator decomposition rather than by independent physics. The result is strongly truncation- and frequency-dependent: Table I gives κ from −5.97% (ℏωi = 10 MeV, Nmax = 2) to −56.92% (ℏωi = 10 MeV, Nmax = 8), and the physically preferred ℏωi ≈ 10–13 MeV shows the largest spread. Section III nevertheless assigns one nucleus-independent κ (the gray band around −15%) to all 15 decays, although the shell-model valence spaces of 10C–74Rb are not the Nmax = 2–8 NCSM spaces used to define κ. No calculation demonstrates that the tritium ratio transfers to medium-mass nuclei; a positive or much smaller κ in those systems would invalidate both the 'δC2 overestimated' claim and the Standard-Model-worsening conclusion. A secondary but real reproducibility issue: the δC1 column in Table I is internally inconsistent with δC = 0.077% unless read as 10× the actual percentage, so the table as printed cannot be used to reproduce the extraction.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the contribution of radial excitations to the Fermi beta-decay matrix element using exact no-core shell model (NCSM) calculations for the mirror decay of tritium. The difference between the converged isospin-symmetry-breaking correction δC and the finite-model-space value δC1 is identified with the radial-overlap correction δC2, which is then computed in a nonorthogonal harmonic-oscillator basis with different oscillator frequencies for initial and final nuclei. The ratio κ = δre_C2/δsm_C2 (radial-excitation contribution divided by radial-diagonal contribution) is extracted as a function of Nmax and ℏω. The paper reports that κ is negative, typically 10–20%, and then uses a nucleus-independent κ in a global fit to the 15 superallowed 0+→0+ transitions, concluding that including radial excitations worsens agreement with the Standard Model under CKM unitarity.","tokens_in":14797,"tokens_out":5641,"duration_ms":57727,"significance":"If the extraction is accepted, the paper would provide the first dedicated numerical estimate of radial-excitation corrections to superallowed Fermi decays, a long-standing question raised by Miller and Schwenk. The use of a convergent NCSM calculation in a nonorthogonal harmonic-oscillator basis is itself a useful proof of principle, and the appendix gives a clear second-quantized formalism for nonorthogonal transition matrix elements. However, the paper's quantitative and global conclusions rest on a model-dependent identification of the NCSM convergence deficit with a radial-overlap correction, and on an untested transfer of a tritium-derived ratio to medium-mass nuclei. These issues are load-bearing for the headline claims.","major_comments":[{"comment":"The global Standard Model test assumes a single nucleus-independent value of κ for all 15 superallowed decays, while Table I reports κ varying from -5.97% to -56.92% depending on Nmax and ℏω_i. No calculation is presented showing that the tritium NCSM ratio transfers to the sd-pf shell-model valence spaces of 10C through 74Rb. If κ is positive or much smaller in magnitude for those nuclei, the conclusion that radial excitations worsen Standard Model agreement does not follow. The authors acknowledge this simplification, but it is load-bearing for the paper's central conclusion and must be either justified or removed.","section":"Sec. III, Eq. (6), Fig. 3"},{"comment":"The identification of the NCSM convergence deficit δC - δC1(Nmax) with the nonorthogonal-basis quantity δC2(Δℏω) is a modeling assumption, not a derivation. Since δC2(Δℏω) can be adjusted through the free parameter Δℏω, the extraction procedure can reproduce any finite-Nmax deficit, and the diagonal-versus-excitation decomposition is then fixed by the harmonic-oscillator structure rather than by independent physics. The statement that the ratio κ is 'expected to be reliable' is not supported by an independent test or a comparison with a realistic radial basis. This directly affects the claim that δC2 values from the shell model are 'likely overestimated.'","section":"Sec. II, Eqs. (2)-(3), Table I"},{"comment":"The δC1 column in Table I is internally inconsistent with the stated converged value δC = 0.077%, unless the tabulated entries are read as 10 times the actual percentages. For example, δC1 = 0.402% at Nmax = 2 and ℏωi = 10 MeV exceeds δC, which would make δC2 = δC - δC1 negative and contradict the positive δC2 = 0.037% listed in the same row. Figure 2 indicates that the plotted δC1 values are scaled by a factor of 10, but the table caption does not state this. As printed, the table cannot be used to reproduce the extraction of δC2 and κ, which are the quantitative basis of the paper.","section":"Table I and Fig. 2"},{"comment":"The central claim that the radial-excitation contribution has 'a typical magnitude of approximately 10% to 20%' of the radial diagonal contribution is not robust across the parameter space shown in Table I. At ℏωi = 10 MeV the extracted κ ranges from -5.97% to -56.92%, with a particularly large jump at Nmax = 8; even at ℏωi = 20–30 MeV the spread is roughly -13% to -18%. No averaging, error bar, or selection criterion is defined that would justify the quoted 10–20% typical range, so the quantitative headline is not supported by the presented data.","section":"Abstract and Sec. II"}],"minor_comments":[{"comment":"The equation for F_t^st appears to have a typesetting error: '2912.95 ± 0.54 |V_ud^st|^2' should read '2912.95 ± 0.54 (s) / |V_ud^st|^2' (or similar), so that the numerical value 3067.26(91) s is obtained.","section":"Eq. (5)"},{"comment":"The axis label 'δC1 [10 %]' is ambiguous; it should state the scaling explicitly, e.g., 'δC1 × 10 [%]' or 'δC1 [10^{-1} %]', and the same convention should be carried into Table I's header.","section":"Fig. 2 and Table I"},{"comment":"The sentence 'The correction values δc1 are red scaled by a factor of 10' contains a typo ('red' should be 'read' or 'are scaled') and should be rewritten for clarity.","section":"Sec. II"},{"comment":"The statement that p- or s-shell nuclei such as tritium are 'free from nodal mixing with core orbits' is imprecise, since tritium has no core; the intended meaning is that the lowest-energy configurations contain only 0s orbitals, so no node-changing overlaps occur among occupied orbitals.","section":"Sec. II"},{"comment":"The remark that three-body forces are 'unlikely to be significant' for radial excitations is plausible but is not tested; a sentence explaining the expected scale of the effect would help the reader judge this assumption.","section":"Sec. II"}],"recommendation":"major_revision","confidential_remarks":"The paper addresses a relevant and debated issue in superallowed beta decay, and the tritium NCSM calculation is a useful contribution. However, the global conclusion depends on an unsupported transfer of a light-nucleus, truncation-dependent ratio to medium-mass nuclei, and Table I has a scaling inconsistency that prevents reproduction of the central extraction. These issues are fixable by a substantial revision that either provides a justification for the transferability or substantially softens the global claims."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read the paper. The genuinely new thing is the first numerical NCSM calculation of the radial excitation piece δre_C2, using a nonorthogonal harmonic-oscillator basis with different frequencies for initial and final nuclei. The formalism is worked out carefully, including the overlap integrals, and they achieve full convergence for tritium, so the extraction of δC from the converged matrix element is solid. The negative sign of the radial excitation contribution is physically plausible and consistent with Miller-Schwenk's schematic result. That is a real step forward.\n\nThe soft spots are where the quantitative claims live. The whole extraction maps the NCSM convergence deficit δC − δC1(Nmax) onto the nonorthogonal-basis δC2(Δℏω). That is a modeling assumption, not a derivation: any finite-Nmax deficit can be fitted by some Δℏω, and the split between δsm_C2 and δre_C2 is then fixed by the harmonic-oscillator decomposition, not by independent physics. The resulting κ is strongly Nmax- and ℏωi-dependent — Table I shows −5.97% to −56.92% at ℏωi=10 MeV — and the paper still asserts a 'typical' −10% to −20% and uses a nucleus-independent κ in the Standard Model test. That assumption is load-bearing and contradicts the paper's own spread.\n\nThere is also a concrete reproducibility problem: the δC1 column in Table I is scaled by 10 relative to the actual percentages (the figure caption says so, but the table does not), so a reader trying to reproduce the extraction from the table will get the wrong deficit. That needs fixing.\n\nOn the Standard Model side, the two-constraint fit prefers κ ≈ +25%, which is opposite to the NCSM sign, and the one-constraint CVC test is basically flat near κ=0. The paper is honest about this inconsistency, but the conclusion that radial excitations worsen Standard Model agreement is only as good as the nucleus-independent κ assumption. For a proof of principle, this is fine; for a quantitative claim, it is not.\n\nBottom line: the paper is worth reading and refereeing. The nonorthogonal-basis method and the sign of the effect are useful contributions. The magnitude and the impact on δC2 are not yet established. A referee should push for uncertainty estimates on κ, tests of the mapping (e.g., three-body forces, heavier nuclei), and removal of the table scaling inconsistency. I would cite it as the first ab initio NCSM estimate, with caution on the numbers.","headline":"First NCSM estimate of radial excitation contribution to Fermi decay, with a plausible negative sign but a load-bearing matching assumption that leaves the magnitude and Standard-Model conclusion shaky.","tokens_in":15297,"tokens_out":3822,"would_cite":true,"duration_ms":34938,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["23.40.-s","21.60.Cs"],"model":"deepseek-v4-flash","headline":"Exact tritium calculations show radial excitations add a negative 10–20% correction to Fermi beta-decay matrix elements, implying shell-model δC2 values are overestimated; including the effect in superallowed decays worsens Standard Model…","keywords":["radial excitations","Fermi beta decay","isospin-symmetry breaking correction","no-core shell model","nonorthogonal harmonic-oscillator basis","superallowed 0+ to 0+ transitions","CKM unitarity test","radial overlap correction"],"falsifier":"Compute the radial-excitation ratio κ for a second superallowed emitter, such as 10C or 14O, with a converged ab initio method or a multi-shell valence-space calculation spanning at least two oscillator shells; if κ turns out positive, or much smaller than 10% of the radial-diagonal term, the claim that shell-model δC2 values are overestimated by 10–20% is refuted. A complementary experimental falsifier: a superallowed Ft measurement with sub-0.01% precision combined with an independent V_ud from neutron or pion decay would show whether the corrected Ft values already agree with the Standard Model without any radial-excitation term.","tokens_in":14192,"feed_emoji":"☢️","tokens_out":12300,"duration_ms":110410,"temperature":0.7,"pith_summary":"This paper asks whether radial excitations—beta-decay transitions in which the single-particle states differ by one radial node—contribute significantly to the isospin-symmetry-breaking correction in Fermi beta decay. Using exact no-core shell model calculations for the tritium mirror decay 3H(β−)3He, with slightly different oscillator frequencies for the mother and daughter nuclei, the authors isolate this contribution and find it negative, about 10% to 20% of the radial-overlap term, and growing with model-space size. They conclude that the δC2 values obtained in the shell-model approach, which omits radial excitations, are likely overestimated. They then fold the effect into the 15 superallowed decays compiled in Ref. [5] and find that a negative radial-excitation term worsens agreement with the Standard Model, while the data would prefer a positive one. The study also serves as a proof of principle for ab initio calculations in a nonorthogonal oscillator basis.","feed_headline":"Radial excitations shrink Fermi beta-decay rate by 10-20%","feed_subtitle":"Tritium calculation says shell-model isospin corrections are overestimated; the effect worsens Standard Model fit.","key_machinery":"The argument carries on a nonorthogonal harmonic-oscillator basis: the initial and final nuclei are described with slightly different oscillator frequencies ℏω_i and ℏω_f, and the beta-decay matrix element is evaluated through radial overlap integrals Ω_{nn'} between oscillator wave functions with different node numbers n and n′. Overlaps with n≠n′ generate the radial-excitation contribution δre_C2; all such overlaps vanish at equal frequencies, so δC2(Δℏω) starts from zero and grows quadratically while the ratio κ stays nearly constant. The NCSM calculation, using an SRG-evolved chiral interaction and a converged model space, provides both the finite-space δC1 and the converged total correction δC, from which κ is extracted by matching δC2 = δC − δC1 against the frequency-shifted calculations. The same decomposition, δC2 = δsm_C2(1+κ), is then applied to the 15-nucleus superallowed dataset.","core_discovery":"The central quantitative result is the ratio κ = δre_C2/δsm_C2, where δsm_C2 is the radial-diagonal (shell-model) overlap correction and δre_C2 is the radial-excitation correction. For 3H(β−)3He in model spaces up to Nmax = 8 with oscillator frequencies between 10 and 30 MeV, the extracted κ ranges from about −6% to −57%, with typical values of −13% to −18% in the more realistic cases, so the radial-excitation contribution is negative and of order 10–20% of the diagonal term. Because the fully converged NCSM matrix element gives δC = 0.077% while the finite-space δC1 values are larger, the missing radial-excitation piece makes the shell-model δC2 an overestimate. When κ is inserted into the 15-nucleus dataset of Ref. [5] under CVC and CKM top-row unitarity constraints, the best fit requires κ ≈ +25%, the opposite sign, so the predicted negative radial excitations drive the corrected Ft values further from the Standard Model prediction; under the CVC-only test the data are insensitive to κ between −30% and +30%.","pith_inferences":["If κ is truly nucleus-dependent, as the paper's own Table I indicates across model spaces, the global single-κ Standard Model test is not decisive; recomputing κ for heavier isotriplets in multi-shell valence spaces and repeating the χ² analysis per nucleus could change the direction of the conclusion.","The same overlap machinery applies to Gamow-Teller decays, where mirror ft asymmetries are used to extract isospin-breaking corrections; radial excitations may contaminate those extractions, and a dedicated calculation would show by how much.","The tension between the negative calculated κ and the positive κ preferred by the CKM-unitarity test could ease if the universal radiative correction or the nuclear-structure-dependent correction in Ref. [5] shifts by more than its quoted uncertainty, so updated electroweak corrections provide a direct check.","Because the Gaussian tails of oscillator functions are unrealistic at large radius, repeating the tritium extraction with a basis that reproduces separation energies would test whether the 10–20% magnitude of δre_C2 survives the change of asymptotics."],"forward_implications":["If the negative radial-excitation contribution is real, shell-model δC2 values for superallowed decays are overestimates, so the true Fermi matrix elements are slightly more suppressed than currently adopted.","The two-constraint Standard Model test (CVC plus CKM top-row unitarity) would need a positive κ of about +25% for consistency, so the predicted negative effect cannot be absorbed by the present radiative and nuclear-structure corrections without changing other inputs.","The CVC-only test, which measures the scatter of corrected Ft values about their mean, cannot discriminate the radial-excitation contribution: its χ²/ν stays below 1 for κ between −30% and +30%.","The nonorthogonal oscillator-basis method is a working proof of principle for ab initio studies with isospin-dependent or state-dependent oscillator frequencies, and it extends to other one-body transition operators."],"supporting_citations":[{"why":"Supplies the experimental ft values, radiative corrections, and shell-model δsm_C2 inputs for the 15 superallowed decays used in the Standard Model tests.","marker":"[5]"},{"why":"The schematic-model study predicting a significant radial-excitation contribution; the paper's negative-κ results are compared with it.","marker":"[50]"},{"why":"The analytical model that derives δC from nodal (radial-excitation) mixing of harmonic-oscillator functions, establishing Z-dependence and shell effects.","marker":"[23]"},{"why":"Defines the shell-model δC2 as a radial mismatch evaluated with realistic radial wave functions, the quantity argued to be overestimated.","marker":"[6]"},{"why":"Provides the chiral N4LO nucleon-nucleon interaction with a 500 MeV regulator from which the no-core shell model Hamiltonian is built.","marker":"[73]"}],"fun_headline_variants":["Radial excitations cut Fermi beta decay by 10-20%","Negative radial excitations shrink beta decay, worsen SM fit","Radial excitations reduce beta rates but worsen CKM unitarity","Tritium beta decay: radial excitations break Standard Model fit","Shell-model beta decay rates overestimated due to radial excitations"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The Standard Model test assumes a single nucleus-independent value of the radial-excitation ratio κ for all 15 superallowed decays, even though the paper's own tritium calculations give κ values ranging from about −6% to −57% depending on model space and oscillator frequency.","fun_headline_variants_meta":{"raw":{"variants":["Radial excitations cut Fermi beta decay by 10-20%","Negative radial excitations shrink beta decay, worsen SM fit","Radial excitations reduce beta rates but worsen CKM unitarity","Tritium beta decay: radial excitations break Standard Model fit","Shell-model beta decay rates overestimated due to radial excitations"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000768,"raw_usage":{"total_tokens":3471,"prompt_tokens":1079,"completion_tokens":2392,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":695,"completion_tokens_details":{"reasoning_tokens":2303}},"tokens_in":695,"tokens_out":2392,"duration_ms":41430,"temperature":1.0,"reasoning_tokens":2303,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T13:52:01.868609+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the radial-excitation ratio κ for a second superallowed emitter, such as 10C or 14O, with a converged ab initio method or a multi-shell valence-space calculation spanning at least two oscillator shells; if κ turns out positive, or much smaller than 10% of the radial-diagonal term, the claim that shell-model δC2 values are overestimated by 10–20% is refuted. A complementary experimental falsifier: a superallowed Ft measurement with sub-0.01% precision combined with an independent V_ud from neutron or pion decay would show whether the corrected Ft values already agree with the Standard Model without any radial-excitation term.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The schematic-model study predicting a significant radial-excitation contribution; the paper's negative-κ results are compared with it."},{"cited_title":"Damgaard, Nucl","cited_arxiv_id":null,"evidence_quote":"The analytical model that derives δC from nodal (radial-excitation) mixing of harmonic-oscillator functions, establishing Z-dependence and shell effects."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the shell-model δC2 as a radial mismatch evaluated with realistic radial wave functions, the quantity argued to be overestimated."}],"review_version":1}