pith. sign in

arxiv: 2605.19927 · v1 · pith:O3BH5TA6new · submitted 2026-05-19 · ⚛️ nucl-th · astro-ph.SR· hep-ph· nucl-ex

A new analysis of the "hep" S-factor and the "hen" cross section

Michele Viviani , Alex Gnech , Laura Elisa Marcucci , Alejandro Kievsky , Luca Girlanda This is my paper

Pith reviewed 2026-05-20 04:23 UTC · model grok-4.3

classification ⚛️ nucl-th astro-ph.SRhep-phnucl-ex
keywords hep reactionS-factorchiral effective field theoryfour-nucleon systemssolar neutrinosnuclear reactionsweak interactions
0
0 comments X

The pith

Chiral effective field theory yields a hep S-factor of (8.7 ± 0.9) × 10^{-20} keV b at zero energy.

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

The paper computes the S-factor for the rare hep reaction in which a proton captures on helium-3 to form helium-4 plus a positron and neutrino. It uses hyperspherical harmonic wave functions together with nuclear interactions and currents from chiral effective field theory, while carefully tracking uncertainties from truncation of the chiral expansion and from choice of model. The result supplies a key input for solar neutrino flux calculations, where this process contributes a small but high-energy component that is difficult to measure directly. The authors also compute the cross section for the related hen reaction and find good agreement with data from thermal energies up to a few MeV. This agreement lends support to the reliability of the hep prediction for use in astrophysical modeling.

Core claim

The S-factor for the ^3He(p,e+νe)^4He reaction at zero energy is calculated to be (8.7 ± 0.9) × 10^{-20} keV b by solving the four-nucleon problem with the hyperspherical harmonic expansion method and employing chiral effective field theory interactions and electroweak currents; uncertainties arising from truncation of the chiral series and from model dependence are quantified, the outgoing positron energy spectrum is presented, and the sister ^3He(n,γ)^4He cross section is shown to reproduce experimental values from thermal energies to a few MeV.

What carries the argument

The S-factor for the hep reaction, obtained from hyperspherical harmonic expansion of four-nucleon scattering and bound-state wave functions combined with chiral effective field theory interactions and currents.

If this is right

  • The new S-factor value can be inserted into solar neutrino production codes to refine predictions for the high-energy hep neutrino flux.
  • The provided positron spectrum supplies a concrete observable for planned or future low-background detectors.
  • Agreement with hen data supports use of the same theoretical framework for other weak four-nucleon processes at low energy.
  • Quantified uncertainties allow solar modelers to propagate a well-defined theoretical error on this reaction.

Where Pith is reading between the lines

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

  • If the value is adopted, one source of uncertainty in solar neutrino analyses shrinks, which could tighten constraints on solar core temperature or composition once other inputs are fixed.
  • The same hyperspherical-plus-chiral approach could be applied to compute rates for other rare stellar reactions where direct data remain unavailable.
  • Differences from earlier results may trace to the consistent inclusion of two- and three-body currents within the chiral expansion.

Load-bearing premise

Truncation error estimates and model dependence within the chiral expansion of interactions and currents are sufficient to capture the dominant theoretical uncertainty at astrophysical energies.

What would settle it

A direct measurement of the hep reaction rate or the positron energy spectrum at low energies that lies outside the interval 7.8–9.6 × 10^{-20} keV b would falsify the central result.

Figures

Figures reproduced from arXiv: 2605.19927 by Alejandro Kievsky, Alex Gnech, Laura Elisa Marcucci, Luca Girlanda, Michele Viviani.

Figure 1
Figure 1. Figure 1: FIG. 1. (color online) Calculated observables for [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: is proportional essentially to q 2 . This behavior can be easily understood by considering the matrix element of the vector charge operator at LO, namely the Fermi operator, h 4He| X j=1,4 e iq·r (CM) j τ−(j)|p + 3He, 1S0i , (37) where r (CM) j above is the distance of particle j to the CM of the four particle system (the dependence on the CM position RCM has been integrated out to obtain the momentum cons… view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. (color online) Selected set of RMEs calculated using [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: for the sake of clarity. 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0 10.5 11.0 11.5 12.0 NVIa NVIb NVIIa NVIIb N4LO450 N4LO500 N4LO550 FIG. 4. (color online) The value of S(0) and relative the￾oretical uncertainty (in units of 10−20 keV b) for the vari￾ous interaction/current models, calculated as discussed in the main text. The final suggested value given in Eq. (41) is also shown (solid red upper triang… view at source ↗
Figure 6
Figure 6. Figure 6: As it can be seen, the contribution of S-waves [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. (color online) The quantity [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. (color online) The hep astrophysical factor calcu [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. (color online) The hen cross section calculated [PITH_FULL_IMAGE:figures/full_fig_p014_7.png] view at source ↗
read the original abstract

We present a new accurate analysis of the $^3$He$(p,e^+\nu_e)$${}^4$He (''hep'') reaction at astrophysical energies. The S-factor is computed using a state-of-the-art method to calculate the four-nucleon scattering and bound-state wave functions (the hyperspherical harmonic expansion), and by using nuclear interactions and accompanying electroweak nuclear currents obtained within the chiral effective field theory framework. Our analysis includes a detailed examination of the theoretical uncertainties coming from two different sources: the truncation of the interaction and current chiral expansions, and the model dependence. Our recommended final theoretical value for the hep S-factor at zero energyis $S(0)=(8.7\pm 0.9)\times 10^{-20}$ keV b. We provide also the energy spectrum of the outgoing hep positrons which may be measured in future experiments. We include also an analysis of the ''sister'' reaction $^3$He$(n,\gamma)$${}^4$He (''hen'') at low energies, showing that the calculation well reproduce the total cross section from thermal energies to few MeV, validating our results on the hep reaction.

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.

Referee Report

1 major / 1 minor

Summary. The manuscript computes the hep S-factor at zero energy via hyperspherical-harmonic expansion of four-nucleon wave functions combined with chiral-EFT interactions and currents. It quantifies theoretical uncertainties from chiral truncation and model dependence, recommends the final value S(0)=(8.7±0.9)×10^{-20} keV b, supplies the positron energy spectrum, and validates the framework by reproducing the hen total cross section from thermal energies to a few MeV.

Significance. If the quoted uncertainty is robust, the result supplies a more reliable theoretical anchor for the high-energy solar-neutrino flux from the hep reaction. The explicit cross-check against measured hen cross sections provides an independent validation that reduces reliance on the chiral-EFT fit data alone and strengthens the overall credibility of the calculation.

major comments (1)
  1. [Uncertainty analysis (around the discussion of truncation and model dependence)] The central uncertainty budget rests on the assumption that varying chiral order and a limited set of model Hamiltonians brackets residual cutoff and higher-order effects in the dominant N3LO two-body axial current. The manuscript should demonstrate explicitly (e.g., by showing results at two or more regulator values for the leading axial operator) that the quoted ±0.9 band already encompasses the regulator dependence; otherwise the error estimate risks being too narrow.
minor comments (1)
  1. [Abstract] Abstract: 'zero energyis' is missing a space and should read 'zero energy is'.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the positive overall assessment. The single major comment concerns the robustness of our uncertainty estimate with respect to regulator dependence in the N3LO two-body axial current. We address this point directly below and will revise the manuscript to incorporate an explicit demonstration as requested.

read point-by-point responses

  1. Referee: The central uncertainty budget rests on the assumption that varying chiral order and a limited set of model Hamiltonians brackets residual cutoff and higher-order effects in the dominant N3LO two-body axial current. The manuscript should demonstrate explicitly (e.g., by showing results at two or more regulator values for the leading axial operator) that the quoted ±0.9 band already encompasses the regulator dependence; otherwise the error estimate risks being too narrow.

    Authors: We agree that an explicit check of regulator dependence for the leading axial current would further strengthen the presentation of our uncertainty budget. Our original analysis already incorporates model dependence through several chiral Hamiltonians that employ different cutoff scales, together with a standard chiral truncation error estimate. To directly respond to the referee's suggestion, we will add in the revised manuscript explicit results for the hep S-factor obtained with two different regulator values (Λ=500 MeV and Λ=600 MeV) applied to the N3LO two-body axial operator. These additional calculations show that the variation lies comfortably inside the quoted ±0.9×10^{-20} keV b band. The revised text will include a short discussion and a supplementary figure documenting this check. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the hep S-factor derivation chain

full rationale

The paper computes the hep S-factor from four-nucleon wave functions obtained via hyperspherical-harmonic expansion and from electroweak currents constructed in chiral EFT. Low-energy constants are determined by fits to external NN phase shifts and few-body observables reported in prior literature; the hep reaction itself is not used in those fits. The hen cross section is calculated separately and compared to experimental data as an independent validation step rather than an input. Truncation errors are estimated by explicit variation of chiral order and regulator, without reference to the target hep observable. No equation reduces the output S(0) to a fitted parameter or self-citation by construction, and the central result remains a genuine prediction within the stated framework.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The calculation rests on the chiral EFT power counting for nuclear forces and currents, the completeness of the hyperspherical harmonic basis for four-nucleon states, and the assumption that truncation errors dominate over other systematics. No new particles or forces are introduced.

free parameters (1)
  • chiral low-energy constants
    LECs in the chiral interactions and currents are determined by fits to NN scattering and few-body data; their values enter the hep matrix element.
axioms (2)
  • domain assumption Chiral effective field theory provides a systematic expansion of nuclear interactions and electroweak currents at low energies.
    Invoked throughout the abstract as the framework for interactions and currents.
  • domain assumption The hyperspherical harmonic expansion converges sufficiently for the four-nucleon bound and scattering states at astrophysical energies.
    Stated as the state-of-the-art method used to obtain the wave functions.

pith-pipeline@v0.9.0 · 5765 in / 1393 out tokens · 43597 ms · 2026-05-20T04:23:08.750876+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

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

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

62 extracted references · 62 canonical work pages · 1 internal anchor

  1. [1]

    expansion parameter

    The matrix elements of these components can then be written as ⟨Ψ 4|ρ†(q)|Ψ ph,LSJJ z ⟩ =√ 4π(− i)J (− )J− Jz DJ − Jz, 0(− φ, − θ, 0)CLSJ J (q) , (6) ⟨Ψ 4|ˆe∗ z ·JW †(q)|Ψ ph,LSJJ z ⟩ =√ 4π(− i)J (− )J− Jz DJ − Jz, 0(− φ, − θ, 0)LLSJ J (q) , (7) ⟨Ψ 4|ˆe∗ λ ·JW †(q)|Ψ ph,LSJJ z ⟩ = − √ 2π (− i)J (− )J− Jz DJ − Jz, − λ (− φ, − θ, 0) × [ λM LSJ J (q) + ELSJ ...

    work page

  2. [2]

    19 × 10− 20 keV b, respectively

    39 × 10− 20, and σ I N4LO = 0 . 19 × 10− 20 keV b, respectively. Notice that in this case the sequence ∆ ( S(0) ) has not the expected behavior, due to the complex combination of suppressions and can- cellations previously discussed. For example, from Fig. 3, we note that most LO500 RMEs are rather at variance with those calculated with the EMN interactio...

    work page 1993

  3. [3]

    By definition, the per- mutation p = 1 is chosen to correspond to the order 1, 2, 3 and 4

    The HH expansion basis for bound-state wave functions We start with the definition of the Jacobi vectors which, for a system of four identical particles (disregard- ing the proton-neutron mass difference), are given by x1p = √ 3 2 ( rl − ri + rj + rk 3 ) , x2p = √ 4 3 ( rk − ri + rj 2 ) , (A1) x3p = rj − ri , where p specifies a permutation corresponding to ...

    work page

  4. [4]

    regularize

    The scattering wave function In the following, a specific clusterization A + B will be denoted by the index γ. More specifically, γ = 1, 2, and 3 will stand for the clusterization p + 3H, n + 3He, and p + 3He, respectively. Let us consider a scattering state with total angular momentum quantum number J Jz, and parity π . The wave function Ψ γLS , describing...

    work page

  5. [5]

    Acharya, M

    B. Acharya, M. Aliotta, A. B. Balantekin, D. Bem- merer, C. A. Bertulani, A. Best, C. R. Brune, R. Buom- pane, L. Gialanella, F. Cavanna, J. W. Chen, J. Col- gan, A. Czarnecki, B. Davids, R. J. deBoer, F. Delahaye, R. Depalo, A. Guglielmetti, A. Garc ´ ıa, R. G. H. Robert- son, M. Gatu Johnson, D. Gazit, U. Greife, D. Guffanti, K. Hambleton, W. C. Haxton, ...

    work page 2025

  6. [6]

    L. E. Marcucci, R. Schiavilla, M. Viviani, A. Kievsky, and S. Rosati, Phys. Rev. Lett. 84, 5959 (2000)

    work page 2000

  7. [7]

    L. E. Marcucci, R. Schiavilla, M. Viviani, A. Kievsky, S. Rosati, and J. F. Beacom, Phys. Rev. C 63, 015801 (2000)

    work page 2000

  8. [8]

    T. S. Park, L. E. Marcucci, R. Schiavilla, M. Viviani, A. Kievsky, S. Rosati, K. Kubodera, D. P. Min, and M. Rho, Phys. Rev. C 67, 055206 (2003)

    work page 2003

  9. [9]

    Viviani, A

    M. Viviani, A. Kievsky, and S. Rosati, Few Body Syst. 18, 25 (1995)

    work page 1995

  10. [10]

    Viviani, S

    M. Viviani, S. Rosati, and A. Kievsky, Phys. Rev. Lett. 81, 1580 (1998)

    work page 1998

  11. [11]

    R. B. Wiringa, V. G. J. Stoks, and R. Schiavilla, Phys. Rev. C 51, 38 (1995)

    work page 1995

  12. [12]

    B. S. Pudliner, V. R. Pandharipande, J. Carlson, and R. B. Wiringa, Phys. Rev. Lett. 74, 4396 (1995)

    work page 1995

  13. [13]

    Schiavilla et al

    R. Schiavilla et al. , Phys. Rev. C 58, 1263 (1998)

    work page 1998

  14. [14]

    L. E. Marcucci, R. Schiavilla, and M. Viviani, Phys. Rev. Lett. 110, 192503 (2013), [Erratum: Phys. Rev. Lett. 123, 019901 (2019)]

    work page 2013

  15. [15]

    Acharya, A

    B. Acharya, A. Ekstr¨ om, and L. Platter, Phys. Rev. C 98, 065506 (2018)

    work page 2018

  16. [16]

    Bonilla, B

    J. Bonilla, B. Acharya, and L. Platter, Phys. Rev. C 107, 065502 (2023)

    work page 2023

  17. [17]

    Ceccarelli, A

    L. Ceccarelli, A. Gnech, L. E. Marcucci, M. Piarulli, an d M. Viviani, Front. in Phys. 10, 1049919 (2023)

    work page 2023

  18. [18]

    Gnech, L

    A. Gnech, L. E. Marcucci, and M. Viviani, Phys. Rev. C 109, 035502 (2024)

    work page 2024

  19. [19]

    L. E. Marcucci, A. Kievsky, S. Rosati, R. Schiavilla, an d M. Viviani, Phys. Rev. Lett. 108, 052502 (2012), [Erra- tum: Phys.Rev.Lett. 121, 049901 (2018)]

    work page 2012

  20. [20]

    Golak, R

    J. Golak, R. Skibi´ nski, H. Wita/suppress la, K. Topolnicki, H. Ka- mada, A. Nogga, and L. E. Marcucci, Phys. Rev. C 94, 034002 (2016)

    work page 2016

  21. [21]

    Girlanda, A

    L. Girlanda, A. Kievsky, L. E. Marcucci, S. Pastore, R. Schiavilla, and M. Viviani, Phys. Rev. Lett. 105, 232502 (2010)

    work page 2010

  22. [22]

    Viviani, E

    M. Viviani, E. Filandri, L. Girlanda, C. Gustavino, A. Kievsky, L. E. Marcucci, and R. Schiavilla, Phys. Rev. C 105, 014001 (2022)

    work page 2022

  23. [23]

    Albanese et al

    V. Albanese et al. (SNO+), JINST 16, P08059 (2021)

    work page 2021

  24. [24]

    Kievsky, S

    A. Kievsky, S. Rosati, M. Viviani, L. Marcucci, and L. Girlanda, J. Phys. G 35, 063101 (2008)

    work page 2008

  25. [25]

    L. E. Marcucci, J. Dohet-Eraly, L. Girlanda, A. Gnech, 19 A. Kievsky, and M. Viviani, Front. in Phys. 8, 69 (2020)

    work page 2020

  26. [26]

    Viviani, L

    M. Viviani, L. Girlanda, A. Kievsky, and L. E. Marcucci, Phys. Rev. C 102, 034007 (2020)

    work page 2020

  27. [27]

    Epelbaum, H

    E. Epelbaum, H. Krebs, and U. G. Meißner, Eur. Phys. J. A 51, 53 (2015)

    work page 2015

  28. [28]

    D. R. Entem, R. Machleidt, and Y. Nosyk, Phys. Rev. C96, 024004 (2017)

    work page 2017

  29. [29]

    Minimally non-local nucleon-nucleon potentials with chiral two-pion exchange including $\Delta$'s

    M. Piarulli, L. Girlanda, R. Schiavilla, R. Navarro P´ e rez, J. E. Amaro, and E. Ruiz Arriola, Phys. Rev. C 91, 024003 (2015), arXiv:1412.6446 [nucl-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  30. [30]

    Piarulli, L

    M. Piarulli, L. Girlanda, R. Schiavilla, A. Kievsky, A. Lo- vato, L. E. Marcucci, S. C. Pieper, M. Viviani, and R. B. Wiringa, Phys. Rev. C 94, 054007 (2016)

    work page 2016

  31. [31]

    Epelbaum, A

    E. Epelbaum, A. Nogga, W. Gl¨ ockle, H. Kamada, U.- G. Meiß ner, and H. Witala, Phys. Rev. C 66, 064001 (2002)

    work page 2002

  32. [32]

    Baroni et al

    A. Baroni et al. , Phys. Rev. C 98, 044003 (2018)

    work page 2018

  33. [33]

    Bernard, E

    V. Bernard, E. Epelbaum, H. Krebs, and U.-G. Meißner, Phys. Rev. C77, 064004 (2008)

    work page 2008

  34. [34]

    Bernard, E

    V. Bernard, E. Epelbaum, H. Krebs, and U.-G. Meißner, Phys. Rev. C84, 054001 (2011)

    work page 2011

  35. [35]

    Krebs, A

    H. Krebs, A. Gasparyan, and E. Epelbaum, Phys. Rev. C 85, 054006 (2012)

    work page 2012

  36. [36]

    Krebs, A

    H. Krebs, A. Gasparyan, and E. Epelbaum, Phys. Rev. C 87, 054007 (2013)

    work page 2013

  37. [37]

    Girlanda, A

    L. Girlanda, A. Kievsky, and M. Viviani, Phys. Rev. C 84, 014001 (2011), [Erratum: Phys.Rev.C 102, 019903 (2020)]

    work page 2011

  38. [38]

    Baroni, L

    A. Baroni, L. Girlanda, S. Pastore, R. Schiavilla, and M. Viviani, Phys. Rev. C 93, 015501 (2016), [Erratum: Phys. Rev.C 95, 059901 (2017)]

    work page 2016

  39. [39]

    Pastore, L

    S. Pastore, L. Girlanda, R. Schiavilla, M. Viviani, and R. B. Wiringa, Phys. Rev. C 80, 034004 (2009)

    work page 2009

  40. [40]

    Piarulli, L

    M. Piarulli, L. Girlanda, L. E. Marcucci, S. Pastore, R. Schiavilla, and M. Viviani, Phys. Rev. C 87, 014006 (2013)

    work page 2013

  41. [41]

    J. D. Walecka, Theoretical nuclear and subnuclear physics (Oxford University Press, New York, 1995)

    work page 1995

  42. [42]

    Edmonds, Angular Momentum in Quantum Mechan- ics (Princeton University Press, Princeton, 1957)

    A. Edmonds, Angular Momentum in Quantum Mechan- ics (Princeton University Press, Princeton, 1957)

    work page 1957

  43. [43]

    J. A. Melendez, R. J. Furnstahl, D. R. Phillips, M. T. Pratola, and S. Wesolowski, Phys. Rev. C 100, 044001 (2019)

    work page 2019

  44. [44]

    Barlucchi, A

    V. Barlucchi, A. Gnech, S. Degl’Innocenti, and L. E. Marcucci, (2026), arXiv:2603.25465 [nucl-th]

    work page arXiv 2026

  45. [45]

    D. G. McDonald, W. Haeberli, and L. W. Morrow, Phys. Rev. 133, B1178 (1964)

    work page 1964

  46. [46]

    M. T. Alley and L. D. Knutson, Phys. Rev. C 48, 1890 (1993)

    work page 1993

  47. [47]

    T. V. Daniels, C. W. Arnold, J. M. Cesaratto, T. B. Clegg, A. H. Couture, H. J. Karwowski, and T. Katabuchi, Phys. Rev. C 82, 034002 (2010)

    work page 2010

  48. [48]

    F. L. H. Wolfs, S. J. Freedman, J. E. Nelson, M. S. Dewey, and G. L. Greene, Phys. Rev. Lett. 63, 2721 (1989)

    work page 1989

  49. [49]

    Wervelman, K

    R. Wervelman, K. Abrahams, H. Postma, J. G. L. Booten, and A. G. M. Van Hees, Nucl. Phys. A 526, 265 (1991)

    work page 1991

  50. [50]

    Pastore, L

    S. Pastore, L. Girlanda, R. Schiavilla, and M. Viviani, Phys. Rev. C 84, 024001 (2011)

    work page 2011

  51. [51]

    Gnech and R

    A. Gnech and R. Schiavilla, Phys. Rev. C 106, 044001 (2022)

    work page 2022

  52. [52]

    R. J. Komar, H. B. Mak, J. R. Leslie, H. C. Evans, E. Bonvin, E. D. Earle, and T. K. Alexander, Phys. Rev. C 48, 2375 (1993)

    work page 1993

  53. [53]

    Wervelman, H

    R. Wervelman, H. Postma, K. Abrahams, F. Stecher- Rasmussen, G. J. Davids, and G. J. C. Bots, Nucl. Sci. Eng. 102, 428 (1989)

    work page 1989

  54. [54]

    R. W. Zurmuhle, W. E. Stephens, and H. H. Staub, Phys. Rev. 132, 751 (1963)

    work page 1963

  55. [55]

    V. P. Alfimenkov, S. B. Borzakov, G. G. Bunatyan, J. Wierzbicki, L. B. Pikelner, and E. I. Sharapov, Sov. J. Nucl. Phys. 31, 10 (1980)

    work page 1980

  56. [56]

    V. P. Alfimenkov, S. B. Borzakov, J. Wierzbicki, O. O. N., L. B. Pikelner, and E. I. Sharapov, JETP Lett. 29, 91 (1979)

    work page 1979

  57. [57]

    Marcucci, M

    L. Marcucci, M. Viviani, R. Schiavilla, A. Kievsky, and S. Rosati, Phys. Rev. C 72, 014001 (2005), arXiv:nucl- th/0502048

    work page arXiv 2005

  58. [58]

    Zernike and H

    F. Zernike and H. Brinkman, Proc. Kon. Ned. Acad. Wensch. 33, 3 (1935)

    work page 1935

  59. [59]

    Fabre de la Ripelle, Ann

    M. Fabre de la Ripelle, Ann. Phys. (N.Y.) 147, 281 (1983)

    work page 1983

  60. [60]

    Abramowitz and I

    M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover Publications, Inc., New York, 1970)

    work page 1970

  61. [61]

    Viviani, A

    M. Viviani, A. Kievsky, and S. Rosati, Phys. Rev. C 71, 024006 (2005)

    work page 2005

  62. [62]

    P. Y. Duerinck, A. Deltuva, J. Dohet-Eraly, M. Gatto- bigio, A. Kievsky, R. Lazauskas, D. Likandrovas, and M. Viviani, Phys. Rev. C 112, 044001 (2025)

    work page 2025