Allowed $\beta$-decay spectrum with numerical electron wave functions

Dong-Liang Fang

arxiv: 1907.04560 · v1 · pith:HFK7GNKWnew · submitted 2019-07-10 · ⚛️ nucl-th · hep-ph

Allowed β-decay spectrum with numerical electron wave functions

Dong-Liang Fang This is my paper

Pith reviewed 2026-05-24 23:37 UTC · model grok-4.3

classification ⚛️ nucl-th hep-ph

keywords beta decayelectron wave functionsFermi functionallowed decayshape factorsnuclear many-body methodsdecay spectraspherical nuclei

0 comments

The pith

Traditional Fermi Function approximations for electron wave functions introduce severe errors into allowed beta-decay spectra.

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

The paper computes allowed beta-decay spectra for spherical nuclei by solving for electron wave functions numerically inside a nuclear many-body framework. It then measures the size of the discrepancies that appear when the same spectra are instead generated from the usual analytical Fermi Function shape factors and from different assumed nuclear charge distributions. The discrepancies turn out to be large enough that the analytical route can distort the simulated spectrum shape by amounts that matter for data analysis. A reader cares because beta spectra enter neutrino-mass searches, reactor monitoring, and astrophysical abundance calculations, so any systematic bias in the shape factor directly affects those applications. The work also supplies a practical route to correct spectra when only limited experimental decay data are available for a given nucleus.

Core claim

Numerical electron wave functions obtained from state-of-the-art nuclear many-body methods serve as a reference standard; when the spectra generated from them are compared with spectra produced by the conventional analytical Fermi Function approximations, the traditional approximations are found to constitute a very severe source of error for spectra simulation.

What carries the argument

Numerical electron wave functions computed inside a nuclear many-body model, used as a benchmark against which analytical Fermi Function shape factors for allowed beta decay are tested.

If this is right

Errors arising from different assumed electric charge distributions inside the nucleus can be quantified separately from the Fermi Function error.
A practical estimation procedure exists that combines the numerical approach with existing experimental decay data for a specific nucleus.
For typical allowed decay channels of spherical nuclei the analytical approximations can distort the entire spectral shape enough to affect simulation results.
The magnitude of the error depends on the specific decay channel and on the nuclear charge distribution chosen for the analytical calculation.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

If the numerical wave functions are trusted, existing beta-spectrum libraries used in reactor or neutrino experiments may need systematic re-evaluation.
The same numerical benchmark could be applied to test whether similar approximation errors appear in forbidden beta decays.
Extending the method to deformed nuclei would test whether the spherical-nucleus results generalize or whether deformation amplifies the discrepancy.
A hybrid scheme that uses the numerical wave functions only near the nucleus and analytic forms farther out could reduce computational cost while retaining accuracy.

Load-bearing premise

The numerical electron wave functions computed with current nuclear many-body methods are accurate enough to serve as a reliable reference standard for the analytical approximations.

What would settle it

High-precision experimental beta spectra measured for the same spherical nuclei whose numerical wave functions were computed would either reproduce the size of the reported discrepancies or show that they are smaller than claimed.

Figures

Figures reproduced from arXiv: 1907.04560 by Dong-Liang Fang.

**Figure 2.** Figure 2: FIG. 2: (Color online) Similar as fig.1 but for decay channels [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3: (Color online) The differential decay width and norma [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4: (Color online) The errors from various approximatio [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗

read the original abstract

Using numerical electron wave functions and state-of-the-art nuclear many-body methods, I evaluate the $\beta$-decay spectra for typical decay channels of spherical nuclei. I check errors brought by various approximations used for deriving the analytical shape factors (the so-called Fermi Function) of allowed decay. I estimate the errors brought by different electric charge distributions and give a way of estimation of $\beta$-spectra with available decay data of specific nuclei. I found that the traditional ways of approximating the electron wave functions by Fermi Function could be a very severe source of error for spectra simulation.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper computes differences between numerical Dirac wave functions and the usual Fermi-function approximation for allowed beta spectra in spherical nuclei, but treats the numerical results as the truth without showing they fit measured spectra better.

read the letter

The main thing here is a set of numerical evaluations of beta-decay spectra for a few spherical nuclei, using Dirac solutions with charge distributions from modern many-body calculations. It then compares those to the analytic Fermi-function shape factors that are standard in the field and reports that the differences can be large. That comparison is the new piece; the method itself is an application of existing numerical techniques rather than a new derivation. The paper also sketches a practical way to adjust spectra when some experimental decay data exist for the same nucleus, which could be useful for people who need quick estimates. The calculations look reproducible in principle since they rely on standard nuclear codes and the Dirac equation. The soft spot is exactly the one the stress-test flagged: the abstract and the claim of “very severe source of error” rest on the assumption that the numerical wave functions are the correct reference. Nothing in the supplied material shows that these numerical spectra agree with measured beta spectra any better than the Fermi-function versions do. Without that check, the differences only demonstrate inconsistency, not which approximation is worse for real data. The work is therefore mainly of interest to specialists who already care about sub-percent spectrum shapes for neutrino or reactor applications and who can judge the numerical methods themselves. It is coherent on its own terms and does not contain obvious internal contradictions, so it is worth sending to referees who can ask for the experimental comparisons and check the numerics. I would not cite it yet, but I would not desk-reject it either.

Referee Report

2 major / 0 minor

Summary. The manuscript evaluates allowed β-decay spectra for spherical nuclei by solving the Dirac equation numerically for electron wave functions, employing nuclear charge distributions obtained from state-of-the-art many-body methods. It quantifies errors introduced by common analytic approximations (Fermi functions) to the shape factor, examines sensitivity to different electric charge distributions, and outlines a procedure for estimating spectra when experimental decay data for specific nuclei are available. The principal conclusion is that traditional Fermi-function approximations can constitute a severe source of error in β-spectra simulations.

Significance. If the numerical spectra are shown to agree better with measured data than the analytic approximations, the work would provide a concrete diagnostic of an important systematic uncertainty in precision β-decay calculations. The explicit use of numerical Dirac solutions together with modern many-body charge distributions is a methodological strength that moves beyond the usual analytic treatment.

major comments (2)

[Abstract] Abstract, paragraph on evaluation of spectra: the central claim that Fermi-function approximations introduce 'very severe' errors rests on the untested premise that the numerical wave functions are the more accurate benchmark. No comparison of either set of spectra to experimental β-decay data is described, so the reported differences demonstrate only internal inconsistency, not which method is closer to reality.
[Abstract] Abstract: the assertion of severe errors is presented without any quantitative magnitudes (e.g., percentage deviations at specific electron energies), without reference to any table or figure that would display the numerical-versus-Fermi differences, and without any statement of how the numerical results were validated against measured spectra. This leaves the primary conclusion unsupported by visible evidence.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below and have revised the abstract to incorporate quantitative details and clearer references while maintaining the focus on the methodological comparison.

read point-by-point responses

Referee: [Abstract] Abstract, paragraph on evaluation of spectra: the central claim that Fermi-function approximations introduce 'very severe' errors rests on the untested premise that the numerical wave functions are the more accurate benchmark. No comparison of either set of spectra to experimental β-decay data is described, so the reported differences demonstrate only internal inconsistency, not which method is closer to reality.

Authors: The numerical wave functions are obtained by directly solving the Dirac equation for realistic nuclear charge distributions from many-body calculations; this constitutes the exact solution within the model. The Fermi function is an analytic approximation to that solution, so the numerical results serve as the benchmark by construction. The manuscript quantifies the resulting discrepancies in the spectra. We agree that experimental comparisons would provide additional support and will add a clarifying statement in the abstract noting that such validation is possible with available decay data but lies outside the present scope, which centers on the size of the approximation errors. revision: partial
Referee: [Abstract] Abstract: the assertion of severe errors is presented without any quantitative magnitudes (e.g., percentage deviations at specific electron energies), without reference to any table or figure that would display the numerical-versus-Fermi differences, and without any statement of how the numerical results were validated against measured spectra. This leaves the primary conclusion unsupported by visible evidence.

Authors: We will revise the abstract to include explicit quantitative examples (e.g., maximum percentage deviations in the spectrum at representative electron energies) and will add direct references to the figures and tables that display the numerical-versus-Fermi comparisons. The numerical results are validated through their origin as solutions of the Dirac equation with state-of-the-art charge distributions; we will insert a brief statement to this effect. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external numerical solutions

full rationale

The paper computes allowed β-decay spectra by solving the Dirac equation for electron wave functions using nuclear charge distributions from state-of-the-art many-body methods, then directly compares the resulting spectra against those obtained from analytic Fermi-function approximations. No load-bearing step reduces a claimed error or spectrum to a fitted parameter, self-definition, or self-citation chain; the numerical reference is treated as an independent input computed from the Dirac equation and external nuclear-structure calculations. The enumerated circularity patterns are absent.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review supplies insufficient detail to enumerate free parameters or invented entities; the central claim rests on the domain assumption that numerical Dirac solutions are more accurate than the Fermi function.

axioms (1)

domain assumption Numerical solutions of the Dirac equation for continuum electron states in a realistic nuclear charge distribution provide a more accurate reference than the analytical Fermi function for allowed beta decays.
Invoked when the paper uses numerical wave functions to check errors in the analytical shape factor.

pith-pipeline@v0.9.0 · 5608 in / 1181 out tokens · 23234 ms · 2026-05-24T23:37:21.780056+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Using numerical electron wave functions and state-of-the-art nuclear many-body methods, I evaluate the β-decay spectrum... traditional ways of approximating the electron wave functions by Fermi Function could be a very severe source of error
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

exact form... λ = G²_β / (2Ji+1) 2π³ ∑_K ∑_κeκν ... ∫ {⟨⟨J_f || (1+gA/gV γ5) T_KLs || J_i⟩⟩ ... ⟨⟨φ_κe(Z) || (1+γ5) T_KLs || φ_κν⟩⟩ r² dr}² p² q² dp

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

28 extracted references · 28 canonical work pages

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43A−1/3 − 6

12A1/3 + 2. 43A−1/3 − 6. 56A−1 fm. And gκ(kR) and fκ(kR) refer to wave functions of electrons with momen- tumk at nuclear surface, hereafter, this refers to surface approximation(SA). The same SA can be applied to neu- trino if we consider neutrino wave functions beyond LA. These approximations simplify the decay width to the form, i. e. with electron SA ...

work page
[2]

[16]), and to facilitate the actual cal- culation, one uses the Fermi Factor for the derivation of the phase space factors as well as spectra[5]

Fermi Factor and point-like charge potential Solution The exact solution of Dirac equations with point-like coulomb potential(Hereafter PCWF) have been obtained analytically ( e.g. [16]), and to facilitate the actual cal- culation, one uses the Fermi Factor for the derivation of the phase space factors as well as spectra[5]. Not so much literature concern...

work page
[3]

beyond Neutrino long wavelength approximation In mostβ -decay calculations, one considers only contri- butions from s-wave neutrino with the long-wave-length approximations, in this work I include contributions also from p-wave and even higher angular momentum. My calculation shows that only s-wave and p-wave with total angular momentum 1/ 2(s1/2 and p1/2...

work page
[4]

The SA can be a reasonable one based on the assumption, that lepton wave functions are nearly constant inside nucleus, this can be true if nu- clear radius is small

Interference between the nuclei and leptons And for point particle potential, I here compare the full numerical nuclear structure calculations with the above surface approximations(SA). The SA can be a reasonable one based on the assumption, that lepton wave functions are nearly constant inside nucleus, this can be true if nu- clear radius is small. On th...

work page
[5]

origin approximations In analytical calculations, people usually use the re- sults at the nuclear surface or at the origin

Surface approximations vs. origin approximations In analytical calculations, people usually use the re- sults at the nuclear surface or at the origin. In this part, I will check which one is better and also which of the choice of nuclear radius helps to solve the problem. Tak- ing a constant value of electron WF(either at the nuclear surface or center) co...

work page
[6]

And in usual calcu- lations, the long-wave length approximation is used and the exact neutrino wave is not well considered

Neutrino wave function At the leading order, the neutrinos propagate in the space with the form of plane wave. And in usual calcu- lations, the long-wave length approximation is used and the exact neutrino wave is not well considered. In above section for the point charge case, for small Q values, the errors are not signiﬁcant. But for large Q values, we ...

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[7]

And with the increased electron energies, this deviation grows larger

Charge distributions Using the electron PCWF, we will get overestimated diﬀerential decay widths, this is due to the fact that such potential shifted the wave functions outwards com- pared to the ﬁnite volume charge potential. And with the increased electron energies, this deviation grows larger. With explicit integrations of PCWF, the intensity of β - 10...

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Huber, Phys

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S. Stoica, M. Mirea, O. Nitescu, J. U. Nabi and M. Ishfaq, Adv. High Energy Phys. 2016, 8729893 (2016)

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[1] [1]

43A−1/3 − 6

12A1/3 + 2. 43A−1/3 − 6. 56A−1 fm. And gκ(kR) and fκ(kR) refer to wave functions of electrons with momen- tumk at nuclear surface, hereafter, this refers to surface approximation(SA). The same SA can be applied to neu- trino if we consider neutrino wave functions beyond LA. These approximations simplify the decay width to the form, i. e. with electron SA ...

work page

[2] [2]

[16]), and to facilitate the actual cal- culation, one uses the Fermi Factor for the derivation of the phase space factors as well as spectra[5]

Fermi Factor and point-like charge potential Solution The exact solution of Dirac equations with point-like coulomb potential(Hereafter PCWF) have been obtained analytically ( e.g. [16]), and to facilitate the actual cal- culation, one uses the Fermi Factor for the derivation of the phase space factors as well as spectra[5]. Not so much literature concern...

work page

[3] [3]

beyond Neutrino long wavelength approximation In mostβ -decay calculations, one considers only contri- butions from s-wave neutrino with the long-wave-length approximations, in this work I include contributions also from p-wave and even higher angular momentum. My calculation shows that only s-wave and p-wave with total angular momentum 1/ 2(s1/2 and p1/2...

work page

[4] [4]

The SA can be a reasonable one based on the assumption, that lepton wave functions are nearly constant inside nucleus, this can be true if nu- clear radius is small

Interference between the nuclei and leptons And for point particle potential, I here compare the full numerical nuclear structure calculations with the above surface approximations(SA). The SA can be a reasonable one based on the assumption, that lepton wave functions are nearly constant inside nucleus, this can be true if nu- clear radius is small. On th...

work page

[5] [5]

origin approximations In analytical calculations, people usually use the re- sults at the nuclear surface or at the origin

Surface approximations vs. origin approximations In analytical calculations, people usually use the re- sults at the nuclear surface or at the origin. In this part, I will check which one is better and also which of the choice of nuclear radius helps to solve the problem. Tak- ing a constant value of electron WF(either at the nuclear surface or center) co...

work page

[6] [6]

And in usual calcu- lations, the long-wave length approximation is used and the exact neutrino wave is not well considered

Neutrino wave function At the leading order, the neutrinos propagate in the space with the form of plane wave. And in usual calcu- lations, the long-wave length approximation is used and the exact neutrino wave is not well considered. In above section for the point charge case, for small Q values, the errors are not signiﬁcant. But for large Q values, we ...

work page

[7] [7]

And with the increased electron energies, this deviation grows larger

Charge distributions Using the electron PCWF, we will get overestimated diﬀerential decay widths, this is due to the fact that such potential shifted the wave functions outwards com- pared to the ﬁnite volume charge potential. And with the increased electron energies, this deviation grows larger. With explicit integrations of PCWF, the intensity of β - 10...

work page

[8] [8]

Huber, Phys

P. Huber, Phys. Rev. C 84, 024617 (2011) Erratum: [Phys. Rev. C 85, 029901 (2012)]

work page 2011

[9] [9]

Mention, M

G. Mention, M. Fechner, T. Lasserre, T. A. Mueller, D. Lhuillier, M. Cribier and A. Letourneau, Phys. Rev. D 83, 073006 (2011)

work page 2011

[10] [10]

T. A. Mueller et al. , Phys. Rev. C 83, 054615 (2011)

work page 2011

[11] [11]

A. A. Sonzogni, T. D. Johnson and E. A. McCutchan, Phys. Rev. C 91, 011301 (2015)

work page 2015

[12] [12]

Fermi, Z

E. Fermi, Z. Phys. 88, 161 (1934)

work page 1934

[13] [13]

H. A. Weidenmuller, Rev. Mod. Phys. 33, 574 (1961)

work page 1961

[14] [14]

Bambynek et al

W. Bambynek et al. , Rev. Mod. Phys. 49, 77 (1977) Er- ratum: [Rev. Mod. Phys. 49, 961 (1977)]

work page 1977

[15] [15]

B¨ uhring, Nucl

W. B¨ uhring, Nucl. Phys. A 40, 472(1963); ibid49, 190(1963)

work page 1963

[16] [16]

Hayen, N

L. Hayen, N. Severijns, K. Bodek, D. Rozpedzik and X. Mougeot, Rev. Mod. Phys. 90, no. 1, 015008 (2018)

work page 2018

[17] [17]

Honma, T

M. Honma, T. Otsuka, B. A. Brown and T. Mizusaki, Phys. Rev. C 65, 061301 (2002)

work page 2002

[18] [18]

B. A. Brown and W. D. M. Rae, Nucl. Data Sheets 120, 115 (2014)

work page 2014

[19] [19]

D. L. Fang, B. A. Brown and T. Suzuki, Phys. Rev. C 88, 034304 (2013)

work page 2013

[20] [20]

Behrens and W

H. Behrens and W. B¨ uhring, Nucl. Phys. A 150, 481 (1970)

work page 1970

[21] [21]

L. R. B. Elton, Nucl. Phys. 5, 173(1958)

work page 1958

[22] [22]

De Vries, C

H. De Vries, C. W. De Jager, C. De Vries, Atom. Data and Nucl. Data Tabl. 36, 495(1987)

work page 1987

[23] [23]

Salvat, J

F. Salvat, J. Fernadez-Varea, and W. Williamson Jr., Comput.Phys.Commun. 90, 151(1995)

work page 1995

[24] [24]

R. C. Young, Phys. Rev. 115, 577(1959)

work page 1959

[25] [25]

Wilkinson, Nucl

D. Wilkinson, Nucl. Phys. Inst. and Meth. A 290, 509 (1990)

work page 1990

[26] [26]

Litvinova, B

E. Litvinova, B. A. Brown, D.-L. Fang, T. Marketin and R. G. T. Zegers, Phys. Lett. B 730, 307 (2014)

work page 2014

[27] [27]

Stoica, M

S. Stoica, M. Mirea, O. Nitescu, J. U. Nabi and M. Ishfaq, Adv. High Energy Phys. 2016, 8729893 (2016)

work page 2016

[28] [28]

D. L. Fang and B. A. Brown, Phys. Rev. C 91, 025503 (2015) Erratum: [Phys. Rev. C 93, 049903 (2016)]

work page 2015