Study of atomic effects on electron spectrum in bound-muon decay process

A. O. Davydov; A. V. Malyshev; M. Y. Kaygorodov; S. B. Zhang; Y. S. Kozhedub; Y. Wu

REVIEW 3 major objections 5 minor 56 references

Study of atomic effects on electron spectrum in bound-muon decay process

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Finite-nuclear-size effects must be included self-consistently in the Dirac equation for both the bound muon and the outgoing electron; for carbon the correction to the electron spectrum exceeds -40% near the endpoint.

desk verdict Careful derivation and a plausible FNS story, but the headline total corrections (+2.5% to +5%) are hard to square with their own FNS-only numbers (-44% for C, -68% for Si); one of those two results is likely wrong. read the letter →

arxiv 2506.02416 v1 pith:ID4ZQPWA submitted 2025-06-03 physics.atom-ph hep-ph

M. Y. Kaygorodov , Y. S. Kozhedub , A. V. Malyshev , A. O. Davydov , Y. Wu , S. B. Zhang This is my paper

classification physics.atom-phhep-ph

keywords bound-muondecayelectronendpointspectrumfinite-nuclear-sizeeffectnucleardeformationvacuumpolarizationscreeningmuon-to-electronconversioncharged-leptonflavorviolation

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 calculates the electron spectrum from bound-muon decay (muon decay in orbit) near its endpoint for carbon, aluminum, and silicon, the target nuclei of upcoming searches for charged-lepton-flavor-violating muon-to-electron conversion. It argues that finite-nuclear-size, nuclear-deformation, vacuum-polarization, electron-screening, and nuclear-recoil corrections must be inserted as local potentials directly into the Dirac equation, so that they change not only the muon binding energy but also the wave functions of the bound muon and the outgoing electron. The central result is that finite-nuclear-size effects dominate near the endpoint: for carbon-12 the correction exceeds -40%, so treating the nucleus as point-like is inadequate even at low $Z$. Including all effects, the total atomic correction near the endpoint is about 2.5% for aluminum, 2.8% for silicon, and up to 5% for carbon. These numbers set the Standard Model background that next-generation muon-to-electron conversion experiments must subtract.

What carries the argument

The load-bearing object is the radial Dirac equation for the bound muon and the unbound electron in the same spherical potential, with each atomic effect added as a local potential: the Fermi nuclear-charge distribution for finite nuclear size, an angle-averaged deformed-Fermi potential for nuclear deformation, Uehling and Wichmann-Kroll potentials for vacuum polarization, an X$\alpha$-type screening potential for electron screening, and a mass-shift operator for the muon recoil. The electron spectrum is evaluated from two equivalent Fermi-theory expressions, a two-particle multipole expansion and an effective one-particle current, whose equivalence generates a definite-integral relation among spherical Bessel functions. The diagnostic that carries the argument is the split of each correction into an energy part, a wave-function part, and the total; this split reveals which effects must be treated self-consistently and which enter only through the endpoint energy.

What would settle it

A high-statistics measurement of the electron spectrum from muon decay in orbit on a carbon target within about 1 MeV of the endpoint could settle the claim, because the self-consistent finite-nuclear-size treatment predicts a pronounced suppression relative to the point-nucleus spectrum, and the observed shape near the endpoint would discriminate between the two. Alternatively, a computation that replaces the $E_e \to E_e - E_e^2/(2M_n)$ substitution with a full relativistic treatment of the final-state electron kinematics would test the load-bearing approximation away from the endpoint.

Watch

Extended reading notes

Core claim

The paper's central claim is that finite-nuclear-size effects on the bound-muon-decay spectrum cannot be captured by a binding-energy shift alone: the finite size of the nucleus must be put into the Dirac equation for both the initial muon and the final electron. With the Fermi nuclear-charge distribution in the potential, the wave-function part of the FNS correction dominates near the endpoint, giving roughly -44% for carbon and -68% for silicon, and the paper concludes that even low-$Z$ nuclei such as $^{12}\mathrm{C}$ require the fully self-consistent treatment. The same analysis shows that nuclear deformation is significant for silicon but not for carbon, that the Uehling vacuum-polarization correction is a few percent, that the Wichmann-Kroll correction is negligible, and that electron screening acts almost entirely through the endpoint energy. As a by-product, two equivalent formulas for the spectrum for an arbitrary bound muon state are derived, and their equality yields a definite-integral relation among products of spherical Bessel functions. The reported total atomic correction near the endpoint is about 2.5% for aluminum, 2.8% for silicon, and up to 5% for carbon.

Load-bearing premise

The load-bearing premise is that the nuclear-recoil correction to the outgoing electron can be modeled by the substitution $E_e \to E_e - E_e^2/(2M_n)$ in the spectrum, which the paper states is valid only near the endpoint where the electron is effectively massless, yet the substitution is applied across the entire plotted range from 100 MeV to the endpoint without a separate uncertainty assigned to it.

Editorial extensions

If this is right

Background spectra for upcoming muon-to-electron conversion searches in carbon, aluminum, and silicon must be computed with finite-nuclear-size potentials in the Dirac equation for both the muon and the electron; point-nucleus results, even for $Z=6$, are not adequate.
For carbon and silicon, the finite-nuclear-size correction is carried mainly by the wave-function modification, so treatments that only shift the muon binding energy will miss the dominant effect.
Nuclear-deformation corrections are appreciable only for silicon (about 0.4-0.5% near the endpoint) and negligible for carbon, so the correction has to be evaluated isotope by isotope.
Electron screening can be accounted for through the muon binding-energy shift alone, because its wave-function contribution is below 0.01% for the nuclei studied.
The total atomic correction near the endpoint, roughly 2.5% for aluminum, 2.8% for silicon, and up to 5% for carbon, must be folded into the estimated background of next-generation charged-lepton-flavor-violation searches.

Reading between the lines

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

For heavier nuclei used in other muon-to-electron conversion searches, such as titanium or gold, the finite-nuclear-size correction grows with $Z$, so the self-consistent treatment advocated here would likely modify the endpoint spectrum even more than in the light-nucleus cases shown.
The equivalence of the two expressions for the spectrum is a general statement about four-fermion contact interactions in a central field, so the derived spherical-Bessel integral relation could transfer to other bound-decay calculations such as bound-beta decay.
The conclusion that screening enters only through the energy shift depends on the $Z-1$ approximation for the electron configuration; a different treatment of the atomic environment around the muon could shift the endpoint and deserves its own uncertainty estimate.
If the kinematic recoil substitution were replaced by a full QED recoil treatment, the reported 2.5-5% total corrections could shift by an amount comparable to the mass-shift uncertainty, which future experiments would need as a theory error bar.

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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

Main thing: the paper has a load-bearing internal inconsistency. Fig. 1 shows the total FNS correction alone is about -44% for carbon near the endpoint and -68% for silicon. Fig. 7 and the abstract claim the combined correction is positive, +2.5% to +5%, in the same energy range. That cannot be right unless the other corrections sum to roughly +49% for C and +70% for Si. The paper's own figures show they do not: Ue gives about +2.2%, SCR is sub-percent at plotted energies, MS is sub-percent, ND is 0.4% at most. The kinematic recoil is consistently included in the baseline and cancels in the relative correction, so it cannot rescue the sign. This is not a misreading; either Fig. 1 or Fig. 7/abstract is wrong, and the paper does not address the tension.

What is genuinely good: the two independent derivations of the electron spectrum (the 2p and 1p approaches) are detailed and appear correct, and the definite-integral relation in Appendix G is a real piece of new formalism. The computed binding-energy corrections for aluminum agree with earlier literature, giving some confidence in the numerical machinery. The physical point that finite nuclear size must be treated self-consistently in the Dirac equation for both muon and electron, even at Z=6, is plausible and important for the CLFV background calculations.

Other soft spots are minor by comparison: spectral corrections carry no propagated uncertainties, no code or numerical tables are released, and the kinematic recoil replacement Ee -> Ee - Ee^2/(2Mn) is applied across the whole plotted range although the paper states it is valid only near the endpoint. Those are addressable.

Who should read this: people working on muon-to-electron conversion backgrounds will want to know about the FNS self-consistency point, but they should not trust the total correction numbers until the contradiction is resolved. The paper deserves a serious referee—the derivations are substantial—but the referee must force the authors to reconcile the sign and magnitude of the headline results. I would not cite it until that is fixed.

Referee Report

3 major / 5 minor

Summary. The paper studies the electron spectrum from bound-muon decay (decay in orbit) near its endpoint, using Fermi effective theory and a central-field relativistic framework. Two formally equivalent expressions for the spectrum are derived, one based on a two-particle spherical-wave treatment and one based on an effective one-particle current, and the equivalence is reduced to an integral identity in Appendix G. Numerical results are presented for 12C, 27Al, and 28Si, with finite-nuclear-size (FNS), nuclear-deformation (ND), Uehling and Wichmann-Kroll vacuum-polarization, electron-screening (SCR), and nuclear-recoil corrections incorporated into the Dirac equation for the muon and the final electron. The paper reports that the FNS correction alone can be about -44% (C) or -68% (Si) near the endpoint, but that the total correction including all atomic effects is about +2.5% (Al), +2.8% (Si), and up to +5% (C).

Significance. If the numerical results are correct, the paper would establish that finite-nuclear-size effects must be treated self-consistently in the Dirac equation for both the bound muon and the outgoing electron in decay-in-orbit background predictions for upcoming muon-to-electron conversion experiments, and it would provide a detailed two-formula derivation of the spectrum. The work uses independent published inputs (nuclear radii, deformation parameters, masses, potential models), does not fit any parameter to the target spectrum, and benchmarks part of the calculation against prior aluminum results, including the Uehling and screening corrections. These are genuine strengths. However, the central quantitative conclusion is currently undermined by an internal inconsistency between the FNS-only corrections and the claimed total corrections, which must be resolved before the results can be relied upon.

major comments (3)

[Section III, Figs. 1 and 7] The total FNS correction shown in Fig. 1 (solid curves, defined by Eq. (15)) is about -44% for 12C and -68% for 28Si near the endpoint, while Fig. 7 reports total corrections that include FNS of about +2.5% (Al), +2.8% (Si), and up to +5% (C). Under the stated definitions, these two figures cannot both be correct: the other displayed corrections (Ue about +2.2%, SCR at most a few percent, ND up to about 0.4%, WK negligible) cannot cancel a -44% to -68% FNS term. The paper must either specify a different reference spectrum for Fig. 7 or correct the numbers; as written, the headline total-correction values in the abstract and conclusion are not supported by Fig. 1.
[Section III, kinematic recoil paragraph] The kinematic nuclear-recoil correction is introduced by replacing Ee with Ee - Ee^2/(2Mn) in Eq. (11) and Eq. (13), and the text states that this approximation is valid only near the spectrum endpoint. The same replacement is nevertheless applied in all plotted results over the full range from 100 MeV to the endpoint, with no separate uncertainty estimate. Since Ee^2/(2Mn) is about 0.5 MeV for carbon at 100 MeV, the approximation error away from the endpoint could be large and energy dependent; the authors should either restrict the plotted range or quantify the uncertainty introduced by this substitution.
[Section III, Eq. (15) and Fig. 7] The combined 'total correction' plotted in Fig. 7 is not defined by an explicit equation; Eq. (15) defines a relative deviation for a single correction. The reference spectrum for the combined case must be stated precisely (for example, point-nucleus with or without kinematic recoil, and with or without the mass-shift recoil correction). The apparent inconsistency between Figs. 1 and 7 may originate in an ambiguous baseline, and specifying the baseline and the composition of the 'nuclear recoil' term is necessary for the results to be reproducible.

minor comments (5)

[Section II, Eq. (5)] The phrase 'unbound sates' should read 'unbound states'.
[Fig. 6 caption] The caption lists 27Al among the isotopes, but the figure legend shows only carbon and silicon; either add the aluminum curve or correct the caption.
[Section III, paragraph after Fig. 7] The phrase 'a total total energy correction' contains a duplicated word and should read 'a total energy correction'.
[Appendix G, Eq. (G6)] The validity of the key integral identity is stated to be 'confirmed numerically', but no precision, convergence criterion, or quadrature parameters are given; adding these details would strengthen the claimed equivalence of Eqs. (9) and (11).
[Section III, screening potential paragraph] The text does not explicitly state whether the electron-screening potential used for the unbound electron is the same as that used for the muon, or whether the Z-1 approximation is applied to both particles; clarifying this would help reproducibility.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the bound-muon decay spectrum is computed from a first-principles derivation with external published inputs, and no fitted parameter or self-citation chain forces the reported predictions.

full rationale

The paper is a forward, first-principles calculation. The electron-spectrum formulas in Eqs. (9) and (11) are derived from the Fermi effective interaction, and the claimed equivalence of the two-particle and effective-one-particle formulations is checked explicitly in Appendix G; no parameter is fitted to the target spectrum or to the reported corrections. The numerical inputs are independent published tables and routines: Fermi nuclear charge distributions from Ref. [30], deformation parameters from Ref. [32], nuclear masses from Ref. [38], vacuum-polarization potentials from the QEDMOD package [33,34], and Xalpha screening potentials [39]. The Xalpha parameter is varied to estimate uncertainty, not tuned to reproduce a desired result. The only self-citation of note is Ref. [31] for the nuclear-deformation potential construction; this is a normal citation to a previously published method, and the ND corrections are at most about 0.5%, so the citation is not load-bearing for the central claims. The decompositions in Eqs. (13)-(15) are bookkeeping definitions that separate energy-shift, wave-function, and total effects; the total correction in Fig. 7 is a direct difference of fully recomputed spectra, so it is not forced by construction. The paper's stated endpoint-only validity of the kinematic recoil replacement, and the possible tension between the large negative FNS correction in Fig. 1 and the positive total correction in Fig. 7, are model-accuracy or consistency issues rather than circularity: they involve no fitted parameter renamed as a prediction and no conclusion that reduces to the paper's own inputs. No circular step can be exhibited.

Assumptions & free parameters 1 free parameters · 8 assumptions · 0 invented entities

The central calculation is a first-principles decay-rate computation in the Fermi effective theory. Its inputs are independent published data (nuclear radii, deformation parameters, masses, QEDMOD potentials) and standard approximations. The only free model element is the Xalpha screening parameter. The unproven numerical identity in Appendix G is a gap in the formal derivation but does not enter the main spectral numerics.

free parameters (1)

Xalpha screening parameter = 0, 2/3, 1 (three variants)
Defines the family of local electron-screening potentials in Sec. III. The authors average results over the three values and use the spread as an uncertainty, so the parameter is a model choice rather than a value fitted to the target spectrum.

assumptions (8)

domain assumption Fermi effective theory with a V-A four-fermion point interaction describes bound-muon decay.
Invoked in Eq. (3) and throughout Sec. II; this is the standard low-energy weak interaction model.
domain assumption Neutrinos are massless, free particles.
Used in the neutrino wave functions and phase-space integrations in Sec. II and Appendix D.
domain assumption The nuclear potential is spherically symmetric and the system is treated in the central-field approximation.
Assumed in Sec. II; deformation is incorporated by angular averaging following Ref. [31].
domain assumption All QED and atomic corrections can be represented as local potentials added to the Dirac equation.
FNS, ND, Ue, WK, and SCR are all inserted as local potentials in Sec. III; this is a standard mean-field treatment.
ad hoc to paper The electron-screening potential is modeled with the Xalpha family and the Z-1 approximation.
Sec. III uses Xalpha = 0, 2/3, and 1 and averages the results; the Z-1 approximation partially accounts for muon-induced shell reconfiguration.
domain assumption The initial-state nuclear-recoil correction is estimated with the non-relativistic mass-shift operator p^2/(2Mn).
Sec. III; the paper cites Ref. [37] for a 2.5 percent accuracy bound and includes the resulting uncertainty in Table I.
domain assumption The final-state electron recoil is described by the substitution Ee -> Ee - Ee^2/(2Mn), valid near the endpoint.
Sec. III; the approximation treats the electron as highly relativistic and massless and is applied over the plotted range.
ad hoc to paper The integral identity Eq. (G6) relating the two spectrum formulas is true.
Appendix G states that the validity of Eq. (G6) is confirmed numerically, not proven analytically; the identity is used to establish equivalence between Eqs. (9) and (11).

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Cite this review

Pith. "Pith review of Study of atomic effects on electron spectrum in bound-muon decay process." pith.science (2026). https://pith.science/paper/ID4ZQPWA

@misc{pith2026250602416,
  author       = {Pith},
  title        = {Pith review of: Study of atomic effects on electron spectrum in bound-muon decay process},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ID4ZQPWA}},
  note         = {Machine review of arXiv:2506.02416}
}

read the original abstract

For the bound-muon decay process, the study of atomic effects on the electron spectrum near its endpoint is performed within the framework of the Fermi effective theory. The analysis takes into account for corrections due to finite-nuclear-size, nuclear-deformation, electron-screening, and vacuum-polarization effects, all of which are incorporated self-consistently into the Dirac equation. Furthermore, the nuclear-recoil correction to the muon binding energy is included. Calculations are carried out for the isotopes of C, Al, and Si, which are of a particular importance for forthcoming experiments aimed at search for the charged-lepton flavor-violating process of muon-to-electron conversion in a nuclear field.

Figures

Figures reproduced from arXiv: 2506.02416 by the authors.

**Figure 3.** FIG. 3. Relative nuclear-deformation corrections to the el [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

**Figure 2.** FIG. 2. Relative wave-function correction due to finite [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: FIG. 4. Relative Uehling corrections to the electron spectr [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]

**Figure 6.** Figure 6: FIG. 6. Relative electron-screening corrections to the ele [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]

**Figure 5.** Figure 5: FIG. 5. Relative Wichmann–Kroll corrections to the electro [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]

**Figure 7.** Figure 7: FIG. 7. Total relative corrections to the electron spec [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]

**Figure 8.** Figure 8: FIG. 8. Electron spectra near the endpoints in the bound [PITH_FULL_IMAGE:figures/full_fig_p008_8.png]

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Reference graph

Works this paper leans on

56 extracted references · 53 canonical work pages

[1]

5 103 103

2 102 102 . 5 103 103 . 5 104 104 . 5 105 28 14Si 12 6C Relative ﬁnite-nuclear-size correction Ee, MeV Carbon (Z = 6): Wave-function correction Energy correction Total correction Silicon (Z = 14): Wave-function correction Energy correction Total correction FIG. 1. Relative ﬁnite-nuclear-size corrections to the ele c- tron spectrum near the endpoint in the...

work page
[2]

takes the following form: V F (1, 2) =GF√ 2δ (⃗ r1 −⃗ r2) × [ γ0γρ ( 1 −γ5)] (1) [ γ0γρ ( 1 −γ5)] (2), (3) where the indices (1) and (2) here and in what follows label the particles the operator acts on, GF is the Fermi constant,γρ are the Dirac gamma matrices, and the sum- mation over the repeated Lorentz indices is implied. The tree-level amplitude of t...

work page
[3]

As a result, the integration domain over the neutrino variables in Eq

ensures that only neutrino states with the appropriate helicity contribute to the transition amplitude. As a result, the integration domain over the neutrino variables in Eq. ( 4) can be formally extended without altering the outcome. By employing diﬀerent representations of the neutrino states in Eq. ( 4), two diﬀerent but formally equivalent approaches ...

work page
[4]

Ardu and G

M. Ardu and G. Pezzullo, Universe 8, 299 (2022)

2022
[5]

A comparison between Eqs

lies in its explicit inclusion of the neutrino wave functions, making it particularly useful for extend- ing the analysis to scenarios which involve hypothetical interactions beyond the Standard Model. A comparison between Eqs. (

work page
[6]

The derivation of this relation, along with a discussion of several special cases, is provided in Appendix G

and ( 11) allows one to derive a deﬁnite- integral relation involving the spherical Bessel functions. The derivation of this relation, along with a discussion of several special cases, is provided in Appendix G. III. NUMERICAL RESULTS We present our results for the electron spectrum in the bound-muon decay process, normalized to the total decay rate of a ...

work page
[7]

Moritsu, Universe 8, 196 (2022)

M. Moritsu, Universe 8, 196 (2022)

work page 2022
[8]

within the central-ﬁeld approximation, while the 3 summation over the total angular-momentum projections is prese nted in Appendix C. The resulting expression for the electron spectrum is given by dW 2p (Ee,n µκ µ) dEe = G2 F 16π ∑ κ eκ ν µ κ ν e l ( Π ljν µ C je 1 2 l0jµ 1 2 C jν e 1 2 l0jν µ 1 2 )2 ∫ Eµ − Ee 0 dEνµ ⏐ ⏐ ⏐ ⏐ ⏐Rs l (eνµ,µν e) + L=l+1∑ L=l−...

work page

Show all 56 references

[9]

( B6) and ( B12), re- spectively

are as follows: Ccγ aαbβ are the Clebsch-Gordan co- eﬃcients and the two-particle radial integrals Rs l (cd,ab ) and Rv Ll (cd,ab ) are deﬁned in Eqs. ( B6) and ( B12), re- spectively. The eﬀective one-particle approach is based on a sep- aration of lepton and neutrino degrees...
[10]

We observe that the − 0. 9 − 0. 8 − 0. 7 − 0. 6 − 0. 5 − 0. 4 − 0. 3 102 102 . 5 103 103 . 5 104 104 . 5 105 12 6C 28 14Si Relative wave-function FNS correction Ee, MeV Carbon (Z = 6): FNS muon, Coulomb electron FNS muon, FNS electron Silicon (Z = 14): FNS muon, Coulomb electr...
[11]

as well as in Eq. ( 13). This approximation is valid near the end point of the electron spectrum, where the momentum transfer to the neutri- nos is minimal and the electron can be treated as a highly relativistic and eﬀectively massless particle. Within this approximation, the...

1937
[12]

5 103 103

007 102 102 . 5 103 103 . 5 104 104 . 5 105 28 14Si 12 6C Relative nuclear-deformation correction Ee, MeV Carbon (Z = 6): Wave-function correction Energy correction Total correction Silicon (Z = 14): Wave-function correction Energy correction Total correction FIG. 3. Relative ...
[13]

5 103 103

05 102 102 . 5 103 103 . 5 104 104 . 5 105 28 14Si 12 6C Relative Uehling correction Ee, MeV Carbon (Z = 6): Wave-function correction Energy correction Total correction Silicon (Z = 14): Wave-function correction Energy correction Total correction FIG. 4. Relative Uehling corre...
[14]

Speciﬁcally, the correction is fully deter- mined by the energy shift, while the wave-function con- tribution is negligible: less than −0.01% for both 12 6C and 28 14Si

In contrast to the FNS and ND corrections, the SCR eﬀect exhibits a completely op- posite trend. Speciﬁcally, the correction is fully deter- mined by the energy shift, while the wave-function con- tribution is negligible: less than −0.01% for both 12 6C and 28 14Si. This indic...
[15]

5 103 103

05 102 102 . 5 103 103 . 5 104 104 . 5 105 28 14Si 12 6C Relative electron-screening correction Ee, MeV Carbon (Z = 6): Wave-function correction (× 100) Energy correction Total correction Silicon (Z = 14): Wave-function correction (× 100) Energy correction Total correction FIG...
[16]

06 100 101 102 103 104 105 δtot(Ee) Ee, MeV Carbon (Z = 6) Aluminium (Z = 13) Silicon (Z = 14) FIG. 7. Total relative corrections to the electron spec- trum near the endpoints in the bound-muon decay process in 12 6C, 27 13Al, and 28 14Si nuclei, incorporating simultaneously t...
[17]

W. J. Marciano, T. Mori, and J. M. Roney, Annual Re- view of Nuclear and Particle Science 58, 315 (2008)

2008
[18]

R. H. Bernstein and P. S. Cooper, Physics Reports 532, 27 (2013)

2013
[19]

Calibbi and G

L. Calibbi and G. Signorelli, La Rivista del Nuovo Ci- mento 41, 71 (2018)

2018
[20]

Bertl, R

W. Bertl, R. Engfer, E. Hermes, G. Kurz, T. Ko- zlowski, J. Kuth, G. Otter, F. Rosenbaum, N. Ryskulov, A. van der Schaaf, et al., The European Physical Journal C - Particles and Fields 47, 337 (2006)

2006
[21]

Teshima, arXiv:1911.07143 [physics]

N. Teshima, arXiv:1911.07143 [physics]

1911 arXiv
[22]

Miscetti, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detec- tors and Associated Equipment 1073, 170257 (2025)

S. Miscetti, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detec- tors and Associated Equipment 1073, 170257 (2025)

2025
[23]

R. M. Dzhilkibaev and V. M. Lobashev, Yadernaya Fizika 49, 622 (1989), [Soviet Journal of Nuclear Physics 49, 384 (1989)]

1989
[24]

R. J. Hill, R. Plestid, and J. Zupan, Physical Review D 109, 035025 (2024)

2024
[25]

Tiomno and J

J. Tiomno and J. A. Wheeler, Reviews of Modern Physics 21, 144 (1949)

1949
[26]

C. E. Porter and H. Primakoﬀ, Physical Review 83, 849 (1951)

1951
[27]

Tenaglia, Il Nuovo Cimento 13, 284 (1959)

L. Tenaglia, Il Nuovo Cimento 13, 284 (1959)

1959
[28]

Gilinsky and J

V. Gilinsky and J. Mathews, Physical Review 120, 1450 (1960)

1960
[29]

¨Uberall, Physical Review 119, 365 (1960)

H. ¨Uberall, Physical Review 119, 365 (1960)

1960
[30]

R. W. Huﬀ, Annals of Physics 16, 288 (1961)

1961
[31]

W. R. Johnson, R. F. O’Connell, and C. J. Mullin, Phys- ical Review 124, 904 (1961)

1961
[32]

H. C. von Baeyer and D. Leiter, Physical Review A 19, 1371 (1979)

1979
[33]

H¨ anggi, R

P. H¨ anggi, R. D. Viollier, U. Raﬀ, and K. Alder, Physics Letters B 51, 119 (1974)

1974
[34]

Herzog and K

F. Herzog and K. Alder, Helvetica Physica Acta 53 (1980)

1980
[35]

Chatterjee and T

L. Chatterjee and T. Roy, Annalen der Physik 492, 321 (1980)

1980
[36]

Shanker, Physical Review D 25, 1847 (1982)

O. Shanker, Physical Review D 25, 1847 (1982)

1982
[37]

Watanabe, M

R. Watanabe, M. Fukui, H. Ohtsubo, and M. Morita, Progress of Theoretical Physics 78, 114 (1987)

1987
[38]

Watanabe, K

R. Watanabe, K. Muto, T. Oda, T. Niwa, H. Ohtsubo, R. Morita, and M. Morita, Atomic Data and Nuclear Data Tables 54, 165 (1993)

1993
[39]

Czarnecki, X

A. Czarnecki, X. Garcia I Tormo, and W. J. Marciano, Physical Review D 84, 013006 (2011)

2011
[40]

Szafron and A

R. Szafron and A. Czarnecki, Physics Letters B 753, 61 (2016)

2016
[41]

Szafron and A

R. Szafron and A. Czarnecki, Physical Review D 94, 051301 (2016)

2016
[42]

Heeck, R

J. Heeck, R. Szafron, and Y. Uesaka, Physical Review D 105, 053006 (2022)

2022
[43]

Salvat, J

F. Salvat, J. M. Fern´ andez-Varea, and W. Williamson, Computer Physics Communications 90, 151 (1995). F. Salvat and J. M. Fern´ andez-Varea, Computer Physics Communications 240, 165 (2019)

1995
[44]

I.Angeli and K.P.Marinova, Atomic Data and Nuclear Data Tables 99, 69 (2013)

2013
[45]

Y. S. Kozhedub, O. V. Andreev, V. M. Shabaev, I. I. Tupitsyn, C. Brandau, C. Kozhuharov, G. Plunien, and T. St¨ ohlker, Physical Review A 77, 032501 (2008)

2008
[46]

Pritychenko, M

B. Pritychenko, M. Birch, B. Singh, and M. Horoi, Atomic Data and Nuclear Data Tables 107, 1 (2016)

2016
[47]

V. M. Shabaev, I. I. Tupitsyn, and V. A. Yerokhin, Phys- ical Review A 88, 012513 (2013)

2013
[48]

V. M. Shabaev, I. I. Tupitsyn, and V. A. Yerokhin, Com- puter Physics Communications 189, 175 (2015). V. M. Shabaev, I. I. Tupitsyn, and V. A. Yerokhin, Com- puter Physics Communications 223, 69 (2018)

2015
[49]

L. W. Fullerton and G. A. Rinker, Physical Review A 13, 1283 (1976)

1976
[50]

A. G. Fainshtein, N. L. Manakov, and A. A. Nekipelov, Journal of Physics B: Atomic, Molecular and Optical Physics 24, 559 (1991)

1991
[51]

V. A. Yerokhin and N. S. Oreshkina, Physical Review A 108, 052824 (2023)

2023
[52]

M. Wang, W. Huang, F. Kondev, G. Audi, and S. Naimi, Chinese Physics C 45, 030003 (2021)

2021
[53]

Sapirstein and K

J. Sapirstein and K. T. Cheng, Physical Review A 66, 042501 (2002)

2002
[54]

Borie and G

E. Borie and G. A. Rinker, Reviews of Modern Physics 54, 67 (1982)

1982
[55]

D. A. Varshalovich, A. N. Moskalev, and V. K. Kherson- skii, Quantum Theory of Angular Momentum (World Scientiﬁc, Singapore, 1988)

1988
[56]

Grant, Advances in Physics 19, 747 (1970)

I. Grant, Advances in Physics 19, 747 (1970). 10 Appendix A: Multipole expansion of the operator V F (1, 2) The multipole expansion of the two-particle interaction operator V F deﬁned in Eq. ( 3) can be derived from the multipole expansion of the delta function δ (⃗ r1 −⃗ r2):...

1970

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

5 103 103

2 102 102 . 5 103 103 . 5 104 104 . 5 105 28 14Si 12 6C Relative ﬁnite-nuclear-size correction Ee, MeV Carbon (Z = 6): Wave-function correction Energy correction Total correction Silicon (Z = 14): Wave-function correction Energy correction Total correction FIG. 1. Relative ﬁnite-nuclear-size corrections to the ele c- tron spectrum near the endpoint in the...

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[2] [2]

takes the following form: V F (1, 2) =GF√ 2δ (⃗ r1 −⃗ r2) × [ γ0γρ ( 1 −γ5)] (1) [ γ0γρ ( 1 −γ5)] (2), (3) where the indices (1) and (2) here and in what follows label the particles the operator acts on, GF is the Fermi constant,γρ are the Dirac gamma matrices, and the sum- mation over the repeated Lorentz indices is implied. The tree-level amplitude of t...

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[3] [3]

As a result, the integration domain over the neutrino variables in Eq

ensures that only neutrino states with the appropriate helicity contribute to the transition amplitude. As a result, the integration domain over the neutrino variables in Eq. ( 4) can be formally extended without altering the outcome. By employing diﬀerent representations of the neutrino states in Eq. ( 4), two diﬀerent but formally equivalent approaches ...

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[4] [4]

Ardu and G

M. Ardu and G. Pezzullo, Universe 8, 299 (2022)

2022

[5] [5]

A comparison between Eqs

lies in its explicit inclusion of the neutrino wave functions, making it particularly useful for extend- ing the analysis to scenarios which involve hypothetical interactions beyond the Standard Model. A comparison between Eqs. (

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[6] [6]

The derivation of this relation, along with a discussion of several special cases, is provided in Appendix G

and ( 11) allows one to derive a deﬁnite- integral relation involving the spherical Bessel functions. The derivation of this relation, along with a discussion of several special cases, is provided in Appendix G. III. NUMERICAL RESULTS We present our results for the electron spectrum in the bound-muon decay process, normalized to the total decay rate of a ...

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[7] [7]

Moritsu, Universe 8, 196 (2022)

M. Moritsu, Universe 8, 196 (2022)

work page 2022

[8] [8]

within the central-ﬁeld approximation, while the 3 summation over the total angular-momentum projections is prese nted in Appendix C. The resulting expression for the electron spectrum is given by dW 2p (Ee,n µκ µ) dEe = G2 F 16π ∑ κ eκ ν µ κ ν e l ( Π ljν µ C je 1 2 l0jµ 1 2 C jν e 1 2 l0jν µ 1 2 )2 ∫ Eµ − Ee 0 dEνµ ⏐ ⏐ ⏐ ⏐ ⏐Rs l (eνµ,µν e) + L=l+1∑ L=l−...

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