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Radial excitations and their potential impact on Fermi $\beta$-decay rates

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read 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…

desk verdict 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. read the letter →

arxiv 2505.20587 v1 pith:EU5HUABV submitted 2025-05-26 nucl-th nucl-ex

L. Xayavong , Y. Lim , N. A. Smirnova , Calvin W. Johnson This is my paper
classification nucl-thnucl-ex PACS 23.40.-s21.60.Cs
keywords radialexcitationsFermibetadecayisospin-symmetrybreakingcorrectionno-coreshellmodelnonorthogonalharmonic-oscillatorbasissuperallowed0+totransitionsCKMunitaritytestoverlap
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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%.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

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.

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 (4)
  1. [Sec. III, Eq. (6), Fig. 3] 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.
  2. [Sec. II, Eqs. (2)-(3), Table I] 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.'
  3. [Table I and Fig. 2] 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.
  4. [Abstract and Sec. II] 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.
minor comments (5)
  1. [Eq. (5)] 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.
  2. [Fig. 2 and Table I] 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.
  3. [Sec. II] 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.
  4. [Sec. II] 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.
  5. [Sec. II] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation found; the paper's central claims rest on model-dependent matching and stated assumptions, not on equations that reduce to their own inputs.

full rationale

The derivation chain is not circular. In Sec. II, δC1 is computed exactly in Nmax-truncated NCSM spaces, and the fully converged value δC = 0.077% is obtained independently at Nmax ≈ 30; the extracted δC2 = δC − δC1 is a bookkeeping definition, not an equation that assumes the conclusion. The nonorthogonal-basis calculation computes δC2(Δℏω) from the same Hamiltonian and overlap integrals, and Table I calibrates Δℏω by projecting the extracted δC2 values onto Fig. 1; this is parameter matching, not a prediction manufactured from the target, because the split into δsm_C2 and δre_C2 is computed from the model and is not fitted to reproduce the claimed negative sign. The sign and magnitude of κ are model outputs with stated Nmax and ℏω_i dependence. Section III does fit κ to the Hardy-Towner ft data via χ2 minimization, but that fit is explicitly presented as a Standard-Model consistency test, and the comparison of the fitted positive κ with the theoretical negative κ is a parameter comparison, not a circular validation. Self-citations to Xayavong-Smirnova frame the δC1 + δC2 decomposition but are not load-bearing for the NCSM calculation. The acknowledged simplification that κ is nucleus-independent (Sec. III) and the unreproducible δC1 scaling in Table I are correctness and reproducibility caveats, not circularity. Therefore no circular step is identified.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The central claim rests on the assumed validity of the nonorthogonal harmonic oscillator basis, the identification of the finite-model-space deficit with δC2, and the extrapolation of a single κ to all superallowed decays. No new particles or forces are introduced.

free parameters (4)
  • κ (nucleus-independent ratio in Standard Model fit) = +25% for [χ²/ν]_2c; 0% for [χ²/ν]_1c
    Sec. III treats κ as a free parameter minimized against experimental Ft values (Eqs. 7 and 8). The best-fit values are not NCSM predictions; they are fitted to the same data used to judge the theoretical expectation.
  • Oscillator frequency ℏω_i = 10, 20, 30 MeV
    Chosen by hand to test convergence; the extracted κ depends strongly on this choice (Table I), so the 'typical' 10-20% range is not robust across parameter choices.
  • Oscillator frequency difference Δℏω = 0.23 to 0.85 MeV
    Determined by projecting the extracted δC2 onto the computed δC2(Δℏω) curves (Sec. II). It is a model-dependent calibration, not a fit to the Standard Model data.
  • SRG flow parameter λSRG = 2 fm^-1
    Chosen by hand as the evolution scale of the chiral interaction. No uncertainty from this choice is quantified.
assumptions (6)
  • domain assumption Chiral N4LO NN interaction with 500 MeV regulator and SRG evolution to λ=2 fm^-1 adequately describes the tritium system; three-body forces are omitted.
    Sec. II footnote 3 acknowledges 3NF enhance binding by >0.5 MeV but argues the effect is unlikely significant for radial excitations. This is asserted, not demonstrated.
  • domain assumption The isospin-symmetric interaction is obtained by switching off Coulomb and averaging T=1 channel matrix elements.
    Sec. II; used for the δC2 calculation.
  • domain assumption The nonorthogonal harmonic oscillator basis with different proton and neutron frequencies preserves translational invariance.
    Sec. II states this without proof; the appendix treats the basis overlap but does not explicitly separate center-of-mass contributions.
  • ad hoc to paper The difference between the finite-Nmax and converged δC1 values equals the radial mismatch δC2 as modeled by Δℏω.
    Sec. II matching procedure; the identification is the core model assumption of the paper.
  • ad hoc to paper κ is nucleus-independent across the 15 superallowed decays.
    Sec. III: 'For simplicity, κ is assumed to be nucleus-independent.' This contradicts the strong Nmax and ℏω dependence found in their own calculation.
  • domain assumption Standard Model inputs from PDG 2018 and Hardy-Towner 2020 (Vus, ΔV_R, δ'_R, δsm_C2) are adopted without re-evaluation.
    Used in Sec. III as the baseline dataset for the consistency tests.

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Cite this review

Pith. "Pith review of Radial excitations and their potential impact on Fermi $\beta$-decay rates." pith.science (2026). https://pith.science/paper/EU5HUABV

@misc{pith2026250520587,
  author       = {Pith},
  title        = {Pith review of: Radial excitations and their potential impact on Fermi $\beta$-decay rates},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EU5HUABV}},
  note         = {Machine review of arXiv:2505.20587}
}
abstract

We investigate the contribution of radial excitations to Fermi $\beta$-decay matrix element. To this end, exact no-core shell model calculations are performed for the mirror $\beta$ decay of tritium, where full convergence can be achieved on an ordinary computer. The differences between the isospin-mixing correction values obtained in the full and in a restricted model spaces are matched to the radial overlap correction term, analogous to that required in the shell-model approach, where the configuration space is extremely limited. We examine this complementary correction term using a nonorthogonal harmonic-oscillator basis, generated by slightly differentiating the oscillator frequencies between the initial and final nuclei, while all desirable properties, including translational invariance, are still preserved. For $N_{\rm max}\le8$, we find that the radial excitation contribution is negative, with a typical magnitude of approximately 10\,\% to 20\,\% of the radial diagonal contribution. This effect becomes more pronounced as the model space increases. Therefore, the $\delta_{C2}$ values obtained in the shell model approach, where radial excitations are not explicitly included, are likely overestimated. Based on experimental $ft$ data and the corrective terms adopted in the survey by Hardy and Towner [Phys. Rev. C {\bf 102}, 045501 (2020)], we show that the incorporation of radial excitations for the superallowed $0^+\rightarrow0^+$ nuclear $\beta$ decay tends however to worsen agreement with the Standard Model.

Figures

Figures reproduced from arXiv: 2505.20587 by the authors.

Figure 1
Figure 1. We remark that, by construction, δC2 is symmet￾ric with respect to ∆ℏω = 0. However, since neutrons in the initial nucleus are more tightly bound than pro￾tons in the final nucleus, it is generally more natural to set ∆ℏω to be slightly larger than 0. Note that we con￾sider only small ∆ℏω; otherwise, Eq. (2), which will be employed in the following, would no longer be justified, as the higher-order terms would becom… view at source ↗
Figure 1
Figure 1. FIG. 1. (Color online) Radial mismatch correction, [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. (Color online) Illustration of the isospin-mixing ( [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figures from the paper (1 more)
Figure 3
Figure 3. Figure 3: (the solid red curve) shows that the shape of the [χ 2/ν] 2c curve is asymmetric with respect to κ = 0, with positive κ values producing better consistency. This is not surprising as the used set of Fti values is known to underestimate the CKM’s top-row unitarity condi…

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. New shell-model calculations of the $\delta_C$ correction to superallowed $0^+\rightarrow0^+$ nuclear $\beta$ decay and standard-model implications

    nucl-th 2025-08 unverdicted novelty 5.0 of 10

    New shell-model calculations update the radial-mismatch correction for superallowed beta decays and yield |V_ud| = 0.97359(33).

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