First-forbidden transitions in the reactor anomaly

Joel Kostensalo; Jouni Suhonen; Leendert Hayen; Nathal Severijns

REVIEW 3 major objections 6 minor 2 cited by

First-forbidden transitions in the reactor anomaly

T0 review · 3 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Microscopic treatment of first-forbidden beta decays yields antineutrino spectral changes of several percent, with a bump in the 4–7 MeV range.

desk verdict A solid shell-model calculation of 36 first-forbidden shape factors that shows a few-percent effect on reactor antineutrino spectra, but the headline claim that it solves both the rate anomaly and the spectral shoulder depends on an uncomputed allowed-shape slope. read the letter →

arxiv 1908.08302 v1 pith:ELXY6GWW submitted 2019-08-22 nucl-th hep-exnucl-ex

Leendert Hayen , Joel Kostensalo , Nathal Severijns , Jouni Suhonen This is my paper · ORCID

classification nucl-thhep-exnucl-ex

keywords first-forbiddenbetadecayreactorantineutrinoanomaly5MeVspectralbumpnuclearshellmodelspectrumshapefactorCoulombcorrectionssummationmethodshoulder

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

Reactor antineutrino spectrum predictions have generally treated beta decays that change nuclear spin or parity as though they had the same spectral shape as fully allowed decays. This paper computes the dominant first-forbidden transitions above 4 MeV with the nuclear shell model and a fully Coulomb-corrected beta-decay formalism, and finds that their energy-dependent shape factors deviate strongly from that approximation. The cumulative antineutrino spectrum develops a bump of up to about 4.5% between roughly 4 and 7 MeV, and up to about 8.5% when the comparison baseline has a steeper allowed slope. Since forbidden transitions carry roughly half the electron flux in the 4–8 MeV window, the paper concludes that neither the reactor flux normalization anomaly nor the so-called 5 MeV spectral bump can be understood without them.

What carries the argument

The load-bearing object is the $\beta$-decay shape factor $C(Z,W)$, the energy-dependent multiplier in the spectrum formula that contains all nuclear-structure information; in the Behrens–Bühring formalism it is built from Coulomb functions such as $\lambda_k$, lepton phase-space factors, and nuclear form factors obtained from shell-model wave functions. The argument works by computing these shape factors fully for 36 dominant first-forbidden transitions instead of setting them to $C=1$ or to the standard weak-magnetism term, and then comparing cumulative electron and antineutrino spectra built with the calculated shapes against spectra built with those allowed approximations. The comparison is what converts a small change in electron spectra into a several-percent change in antineutrino spectra, because the steeply falling cumulative flux amplifies any reweighting of the high-energy tail.

What would settle it

A high-statistics measurement of the beta spectrum of a dominant fission fragment such as 92Rb or 96Y, converted into an extracted shape factor, would settle the claim: if the measured shape factor is flat instead of the calculated downward-sloping curve, the predicted several-percent antineutrino bump shrinks or disappears.

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Extended reading notes

Core claim

On its own terms, the paper establishes that first-forbidden transitions are not a small correction in reactor antineutrino spectra but a dominant component above 4 MeV, and that their true shapes change the predicted flux in exactly the region where experiments see an excess. For 36 transitions selected by their fission yield and branching ratio, the paper evaluates the full shape factor in the Behrens–Bühring formalism without the usual $\xi$-approximation, including Coulomb distortion of the electron wave function and finite-size effects. The calculated shape factors for many pseudovector transitions slope downward with energy, while some pseudoscalar transitions show strong quadratic behaviour from higher-order operators, and unique transitions require the Coulomb function $\lambda_2$ for percent-level accuracy. When these shapes are folded into summation and composite spectra matched to measured electron data, the electron spectrum changes by only about 1–2%, but the antineutrino spectrum rises by several percent in the 4–7 MeV window, with the exact magnitude depending on the allowed shape to which the comparison is made.

Load-bearing premise

The central quantitative claim assumes that the 'allowed' spectral shape used as the comparison is the right baseline, even though the paper itself lists missing corrections to that baseline; a different allowed slope would change both the size and the shape of the predicted bump.

Editorial extensions

If this is right

Spectral-shape analyses that assume forbidden decays are allowed will mis-estimate the high-energy antineutrino flux by several percent, so the 5 MeV excess cannot be interpreted without a forbidden-transition correction.
The forbidden-transition contribution adds a correlated spectral uncertainty comparable to or larger than the leading systematic uncertainties of the standard summation model, so previous significance estimates for the rate anomaly were missing a major error term.
With the 36 numerical shape factors alone, the predicted inverse-beta-decay rate rises by about 0.8(5)%; including the parametrized shapes for all other forbidden branches raises the shift to about 2.3(13)%, which shifts but does not remove the anomaly.
If the true average slope of allowed shapes is near 2.4% per MeV rather than the standard 0.67% per MeV, the forbidden-transition bump reaches the size of the observed shoulder, offering a single mechanism for both the rate and shape discrepancies.
Because the effect is nearly identical for the main fission actinides, small differences in reactor fuel composition do not change the predicted distortion, making the correction robust for current and near-future experiments.

Reading between the lines

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

A direct shell-model survey of the allowed-shape slopes for the dominant fission fragments would decide whether the 2.4%/MeV baseline is physical; the paper motivates it but does not compute it.
High-statistics beta-spectrum measurements of individual dominant fragments (for example 92Rb or 96Y) could extract their shape factors and anchor or falsify the parametrization before it is applied to reactor data.
The heavy-tailed, multimodal parameter distributions suggest that older Gaussian-style uncertainty propagation for forbidden transitions underestimates the high-energy spectral error; the Monte Carlo approach used here may be the safer template for future summation calculations.
If sterile-neutrino searches continue to rely on reactor spectral predictions, the size of this forbidden-transition uncertainty will matter as much as detector systematics in the 4–7 MeV region.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Two things to know up front. This is the most complete microscopic treatment of first-forbidden transitions in reactor antineutrino spectra so far, with 36 shell-model shape factors and full Behrens-Bühring Coulomb corrections. And the paper's biggest claim—that these forbidden transitions, combined with a steeper allowed shape slope, can resolve both the reactor normalization anomaly and the 5 MeV shoulder—is a proposal, not a demonstrated result.

What's genuinely new: the extension from 29 to 36 transitions, the explicit shape factors for a large fraction of the flux above 4 MeV, and the Monte Carlo parametrization that turns the scatter in those shape factors into a correlated spectral uncertainty. The 140Xe comparison to Fang and Brown is favorable, which is a good sanity check. The point that forbidden transitions dominate the flux in the 4–8 MeV window is well made and is the real takeaway.

The soft spots are mostly ones the authors flag themselves. The forbidden-only bump reaches about 4.5% relative to the weak-magnetism slope of 0.67%/MeV; the advertised ~8.5% bump comes from replacing that with an ad hoc 2.4%/MeV slope. The paper lists missing induced-tensor and Lambda-prime corrections in Sec. IID and says they will be investigated later, so the combined-solution conclusion rests on a quantity that is not computed. Half-lives for 86Br and 89Br are not reproduced, the parametrization is validated only in-sample, the 2–4 MeV increase is acknowledged as possibly an artifact, and the uncertainty band is a bound treated as 1 sigma. Those are real limitations, but they do not undercut the basic finding that forbidden shapes deviate strongly from allowed approximations and affect antineutrino spectra by several percent.

The citation pattern is appropriate; this builds on earlier work by the same group and by Fang and Brown, and the formalism references are standard. The paper is written honestly, with limitations stated rather than hidden.

Bottom line: this is a useful paper for anyone working on reactor flux predictions or sterile neutrino fits. It deserves peer review. A referee should push on whether the allowed shape slope can be pinned down, because that controls the magnitude of the effect and the headline conclusion. I would send it out.

Referee Report

3 major / 6 minor

Summary. The paper presents microscopic shell-model calculations of the dominant first-forbidden beta transitions contributing to reactor antineutrino spectra above 4 MeV, using the Behrens-Bühring formalism with full Coulomb corrections. The authors calculate shape factors for 36 transitions, find strong deviations from the allowed approximation, and use these to construct both a direct numerical correction and a Monte Carlo parametrization covering the remaining forbidden branches in summation calculations. They then compare the resulting electron and antineutrino spectra with the ILL data and with the Huber-Mueller model, concluding that a correct treatment of forbidden transitions is indispensable for both the normalization anomaly and the spectral shoulder, and proposing that a larger allowed-shape slope (2.4%/MeV) could resolve both simultaneously when combined with their forbidden corrections.

Significance. If the main results hold, this is a significant contribution to the reactor antineutrino problem. The work applies a standard and sophisticated formalism (Behrens-Bühring with shell-model matrix elements) to a much larger set of forbidden transitions than previous microscopic studies, and it provides a transparent uncertainty treatment by varying gA and epsilon_MEC. The comparison with the independent ILL electron data through a composite approach is a strength, as is the favorable reproduction of the 140Xe shape factor against Fang and Brown. The paper also ships a detailed parametrization procedure and Monte Carlo code logic, which is useful for future summation calculations. However, the central quantitative claim about simultaneously solving the anomaly and shoulder depends on an uncomputed baseline slope for allowed transitions, and the paper itself identifies several transitions (86Br, 89Br, 136Te) where the calculations are not fully validated. The significance of the paper is therefore real but conditional on resolving these load-bearing uncertainties.

major comments (3)

[Sec. VII B 1, Fig. 15, and Sec. II D] The claim that a proper treatment of forbidden transitions can simultaneously solve the normalization anomaly and spectral shoulder rests on the 2.4%/MeV slope of the allowed shape factor, which is never computed. In Sec. II D the authors list the missing induced-tensor term (Eq. 23), the full weak-magnetism expression (Eq. 22), and the Lambda-prime finite-size correction (Eq. 24), and state that these will be investigated in future work. The forbidden-only bump in Fig. 14 reaches only about 4.5% when compared with the simplified 0.67%/MeV weak-magnetism slope, whereas Fig. 15 shows an 8.5% bump only after replacing that baseline with an ad hoc 2.4%/MeV slope. Since the true yield-weighted average allowed slope in the 4-7 MeV region is not computed, the central conclusion in Sec. IX that an increased allowed slope 'can yield a solution' is a proposal rather than a demonstrated result. This is the most load-bearing quantitative assertion of the paper and needs to be either supported by an explicit calculation of the allowed shape factors for the dominant allowed branches or softened accordingly.
[Sec. IV C, IV D, and IV E; Table I] The reliability of the numerical shape factors is not uniform across the 36 transitions, and the paper acknowledges unresolved discrepancies. The half-lives of 86Br and 89Br are not reproduced even with the larger jj45pna model space (Sec. IV C), and the 136Te shape-factor slope is sensitive to the phase convention and to the gA/gV ratio (Sec. IV E). Since 86Br (Delta J = 1) and 89Br contribute to the highest-energy part of the spectrum and show some of the largest deviations in Fig. 3, these unresolved cases could disproportionately affect the magnitude of the bump in the 4-7 MeV region. The paper should quantify how the final cumulative spectral changes in Figs. 14 and 16 would shift if the problematic transitions were assigned conservative alternative shape factors (e.g., the allowed shape or the unique-first-forbidden shape), or should explicitly state which transitions dominate the bump and show that they are among the validated cases.
[Sec. VI B, Fig. 10, and Sec. VII B 2] The parametrized shape-factor distributions are validated against the same numerical shape factors to which the polynomial of Eq. (26) was fitted, so Fig. 10 is not an independent test of the parametrization's predictive power. The paper then applies this parametrization to roughly 1600 forbidden branches in the database, assuming the 36 selected transitions are representative, an assumption stated explicitly in Sec. VI B 1. The authors themselves note in Sec. VII B 2 that the 2-4 MeV enhancement in the parametrized results 'could be a true verifiable feature or a limitation of our parametrization' and in Sec. VIII A that the increased spread in shape factors 'is possibly a limitation of our current approach.' Given that the parametrization uncertainty is comparable to or larger than the main systematic uncertainties in the Huber-Mueller model (Appendix), the paper should provide a more direct test of representativeness, for example by using a leave-one-out cross-validation on the 36 transitions or by comparing the parametrized shape-factor distributions to the few available experimental shape-factor measurements beyond half-lives.

minor comments (6)

[Abstract and Sec. IV C] The abstract uses the phrase 'ab initio electron cumulative spectra,' but the calculations are shell-model calculations with effective interactions, not ab initio; please rephrase to avoid confusion.
[Sec. IV C] The sentence 'This has for many isotopes resulted in a correction of branching ratios to high-lying states which had previously gone eluded due to the pandemonium effect' contains a typo; 'gone eluded' should be 'escaped detection' or 'been missed.'
[Sec. II C 2, Eq. (18)] The symbol for antineutrino momentum in Eq. (18) is written as pν, but elsewhere the paper uses p with W0 - W; please define pν and p_e explicitly in the text preceding Eq. (18) for clarity.
[Table III and Sec. VII C] In Table III, the row labels φ and R_IBD are not defined in the caption or in the text before the table; please define them as the integrated antineutrino flux and the inverse-beta-decay-weighted flux, respectively.
[Sec. VII A and Fig. 11] The caption of Fig. 11 mentions 'ENDF Q' but the legend in the figure shows 'ENDF Q' and the text refers to the Qβ approximation; please make the label consistent and explain the acronym in the caption.
[Sec. VIII A, Fig. 18] The figure caption labels the curves as '0.67%' and '2.4%' but does not state that these are slopes in units of %/MeV; please specify the units explicitly.

Circularity Check

0 steps flagged · score 3.0 of 10

Central forbidden-bump calculation is independent; only in-sample parametrization validation and a speculative allowed-slope extrapolation limit the combined-solution claim.

full rationale

The paper's central numerical result—the 4-5% antineutrino bump from the 36 explicitly calculated shell-model shape factors (Secs. IV and VII B 1, Figs. 5 and 14)—is not circular. The shape factors are computed in the Behrens-Bühring formalism with shell-model wave functions, calibrated only to external half-lives through gA and εMEC, and the cumulative electron spectra are matched to the independent ILL electron data via virtual branches; the antineutrino result is therefore an inversion rather than a fit to the observed bump. No load-bearing self-citation chain is present: the earlier paper [18] is the same group's shorter report of this calculation, and the formalism itself cites standard external references. The two genuine weaknesses are (i) the in-sample parametrization validation flagged above, which affects the Monte Carlo uncertainty and the 2-4 MeV parametrized features but not the main 4.5% forbidden bump, and (ii) the combined rate-plus-shoulder solution in Secs. VII B 1 and IX, which depends on an uncomputed 2.4%/MeV allowed-slope baseline and is explicitly described as a proposal 'currently under investigation.' These are limitations and extrapolations rather than reductions of the central derivation to its own inputs, so the overall circularity score is low.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The paper introduces no new particles, forces, or conserved quantities. Its central numerical results rest on standard beta-decay formalism, shell-model Hamiltonians, and tuned coupling constants; the main extrapolative leap is the parametrization from 36 computed shape factors to the full forbidden-transition database, plus the composite fit to ILL data.

free parameters (5)

gA (effective axial coupling) = 0.9 central; varied 0.7, 0.9, 1.0, 1.27
Quenched axial vector coupling tuned to reproduce experimental half-lives of the calculated transitions; the spread in final spectra from this variation is treated as the uncertainty.
epsilon_MEC (axial charge meson-exchange enhancement) = 1.4 central; varied 1.4, 1.7, 2.0
Enhancement factor for the pseudoscalar (gamma5) operator, fitted to experimental half-lives of the pseudoscalar transitions.
gV quenching for pure Delta J = 1 transitions = not specified
The authors state that both gA and gV need to be quenched to reproduce half-lives for pure Delta J = 1 transitions (Sec. IV C), but the exact gV values are not given.
Parametrization coefficients a, b, c (Eq. 26) = Delta J=0: a ~ 0.01, b ~ 0.00, c ~ 0.00; Delta J=1: a ~ 0.07, b ~ 0.10, c ~ 0.00
Polynomial fits to the 36 numerically calculated shape factors; the fitted distributions are sampled and applied to all other forbidden branches in the Monte Carlo summation.
Virtual branch parameters (average Z, endpoint energies) = randomized within fit uncertainties
Used in the composite approach to force agreement with ILL electron spectra; the conversion to antineutrino spectra depends on these fitted branches.

assumptions (6)

domain assumption The Behrens-Buhring formalism with the impulse approximation and a uniformly charged sphere describes beta-decay shape factors accurately.
Used throughout Sec. II to express shape factors in terms of nuclear matrix elements and Coulomb functions; standard in the field but an approximate framework.
domain assumption The shell-model effective Hamiltonians (glepn, jj45pna, jj56pnb) describe the relevant nuclear states for all 36 transitions.
The calculations rely on these interactions; the paper reports good half-life agreement for most cases but explicitly notes failure for 86Br and 89Br (Sec. IV C).
ad hoc to paper The 36 selected transitions are a representative sample of all ~1600 forbidden branches in the region of interest.
Needed for the parametrization extension to the full database; the authors acknowledge this is a limitation, writing that the 2-4 MeV feature 'whether this is a true verifiable feature or a limitation of our parametrization remains to be seen' (Sec. VII B 2).
ad hoc to paper The allowed reference shape factors (C=1 or weak magnetism 0.67%/MeV) are adequate baselines against which forbidden effects are measured.
The magnitude of the forbidden-induced bump depends on this baseline; Sec. II D lists missing allowed corrections (induced tensor, Lambda-prime terms) and Fig. 15 shows the bump grows to 8.5% with a 2.4%/MeV allowed slope.
domain assumption Screening corrections to the Coulomb functions are negligible for cumulative reactor spectra.
Explicitly omitted in Sec. II C 2 with the expectation that they do not contribute substantially in cumulative beta spectra; no quantitative bound is given.
ad hoc to paper The residual electron flux between the summation model and ILL data can be represented by a small number of virtual branches.
The composite approach in Sec. VII A requires fitting virtual branches to the ILL electron spectra; the resulting antineutrino bump is sensitive to the shapes assigned to these branches.

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Cite this review

Pith. "Pith review of First-forbidden transitions in the reactor anomaly." pith.science (2026). https://pith.science/paper/ELXY6GWW

@misc{pith2026190808302,
  author       = {Pith},
  title        = {Pith review of: First-forbidden transitions in the reactor anomaly},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ELXY6GWW}},
  note         = {Machine review of arXiv:1908.08302}
}

read the original abstract

We describe here microscopic calculations performed on the dominant forbidden transitions in reactor antineutrino spectra above 4 MeV using the nuclear shell model. By taking into account Coulomb corrections in the most complete way, we calculate the shape factor with the highest fidelity and show strong deviations from allowed approximations and previously published results. Despite small differences in the ab initio electron cumulative spectra, large differences on the order of several percents are found in the antineutrino spectra. Based on the behaviour of the numerically calculated shape factors we propose a parametrization of forbidden spectra. Using Monte Carlo techniques we derive an estimated spectral correction and uncertainty due to forbidden transitions. We establish the dominance and importance of forbidden transitions in both the reactor anomaly and spectral shoulder analysis. Based on these results, we conclude that a correct treatment of forbidden transitions is indispensable in both the normalization anomaly and spectral shoulder.

Figures

Figures reproduced from arXiv: 1908.08302 by the authors.

**Figure 2.** shows the contribution of the latter relative to the measured spectra at the ILL for 235U [47]. 10 9 10 6 10 3 10 0 Spectrum [a.u.] ILL 2000 4000 6000 8000 10000 Energy [keV] 0 20 40 Contribution [%] [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

**Figure 3.** Calculated shape factors C for the 36 first-forbidden transitions in Table I versus electron kinetic energy, categorized according the spin-parity change of the transition. For allowed transitions C ≈ 1, represented by the black dotted line. Each shape factor was normalized to its value at E = 0. Results correspond to gA = 0.9 and MEC = 1.4, where applicable [18, 54]. Note the difference in scales on the y-axis. … view at source ↗

Figures from the paper (14 more)

**Figure 4.** Figure 4: Numerical shape factors calculated with the nuclear [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]

**Figure 5.** Figure 5: Top panel: Change in the predicted partial electron [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]

**Figure 6.** Figure 6: Top (bottom): Uncertainty in the relative change [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]

**Figure 7.** Figure 7: Overview of the spectral composition of the cu [PITH_FULL_IMAGE:figures/full_fig_p013_7.png]

**Figure 8.** Figure 8: Distribution of fit parameters a, b and c (Eq. (26)) from the numerical results of Sec. IV for pseudoscalar and pseudovector transitions. Full lines represent an expectation of the underlying distribution using Gaussian kernel density estimation. This corresponds to th…

**Figure 10.** Figure 10: Assessment of the quality of parametrized shape [PITH_FULL_IMAGE:figures/full_fig_p015_10.png]

**Figure 9.** Figure 9: Distribution of fit parameters a, b and c (Eq. (26)) and their correlation projections from the numerical results of Sec. IV for pseudoscalar (top) and pseudovector (bottom) transitions. The appearance of heavy tails and multimodal distributions show the need for the i…

**Figure 11.** Figure 11: Comparison of different ways of combining the [PITH_FULL_IMAGE:figures/full_fig_p016_11.png]

**Figure 12.** Figure 12: Comparison of the summation and composite ap [PITH_FULL_IMAGE:figures/full_fig_p017_12.png]

**Figure 13.** Figure 13: Relative change in the cumulative electron spectra [PITH_FULL_IMAGE:figures/full_fig_p018_13.png]

**Figure 14.** Figure 14: Relative change to the cumulative antineutrino [PITH_FULL_IMAGE:figures/full_fig_p018_14.png]

**Figure 16.** Figure 16: Relative change to the cumulative antineutrino [PITH_FULL_IMAGE:figures/full_fig_p019_16.png]

**Figure 18.** Figure 18: Comparison of the Daya Bay shape discrepancy [PITH_FULL_IMAGE:figures/full_fig_p021_18.png]

**Figure 17.** Figure 17: Comparison of the expected spectrum change due [PITH_FULL_IMAGE:figures/full_fig_p021_17.png]

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Parametrized forbidden shape factors The expected shape factor contribution from pseu- doscalar operators is approximately equal to unity (see, e.g. Eq. (7)), so that nearly all of the deviations observed in Fig. 3 arise from higher-order operators contributing to the Jπ→ J−π transition. Depending on the sign of these contributions one arrives at a positi...

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

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Parametrized forbidden shape factors The expected shape factor contribution from pseu- doscalar operators is approximately equal to unity (see, e.g. Eq. (7)), so that nearly all of the deviations observed in Fig. 3 arise from higher-order operators contributing to the Jπ→ J−π transition. Depending on the sign of these contributions one arrives at a positi...

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Monte Carlo procedure The jump to a generalized summation calculation tak- ing into account all ﬁrst forbidden transitions is now straightforward. The summation calculations proceed as normal, with the exception of forbidden transitions. Here, the spin-parity change is determined and the corre- sponding approximate shape factor is taken as described in th...

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IV into the summation and composite calculations

Numerical shape factors The central result of this work is the direct eﬀect of in- cluding the numerically calculated shape factors of Sec. IV into the summation and composite calculations. For the former, the results were already demonstrated in Fig. 5, where the shaded areas correspond to the total diﬀer- ence in both electron and antineutrino spectra. ...

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VI, we go one step further and use the parametrization derived there to look at the additional eﬀects of including all other known for- bidden transitions in a stochastic way

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