Pith. sign in

REVIEW 4 major objections 4 minor 5 cited by

Effects of beyond-mean-field correlations on nuclear Schiff moments

T0 review · 4 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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

Pith's one-line read The first beyond-mean-field calculation of nuclear Schiff moments finds that shape mixing and symmetry restoration significantly change them, cutting the octupole-enhanced moment of 225Ra by up to a factor of three.

desk verdict First MR-CDFT calculation of Schiff moments; Ra225 is credible, but Xe/Hg numbers rest on an admitted truncation of the intermediate-state space. read the letter →

arxiv 2507.01369 v1 pith:52R7O2E5 submitted 2025-07-02 nucl-th hep-phphysics.atom-ph

E. F. Zhou , J. M. Yao , J. Engel , J. Meng This is my paper
classification nucl-thhep-phphysics.atom-ph MSC 81V35
keywords Schiffmomentbeyond-mean-fieldgeneratorcoordinatemethodcovariantdensityfunctionaltheoryelectricdipoleoctupoledeformationCPviolationnuclearstructure
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

The paper presents the first calculation of nuclear Schiff moments that goes beyond the mean-field approximation, using multireference covariant density functional theory with generator-coordinate mixing of shapes and restoration of parity, particle number, and angular momentum. It claims that these beyond-mean-field correlations modify the moments substantially, not only in soft nuclei like 129Xe and 199Hg but also in the well-deformed 225Ra, where shape mixing reduces the octupole-enhanced structure factors by up to a factor of three. If the claim holds, nuclear-structure uncertainty in interpreting electric-dipole-moment experiments is larger than mean-field estimates suggest, and the octupole enhancement of the 225Ra Schiff moment is weaker than previously thought. The paper also reports a correlation between the contributions of intermediate states to the moment and their B(E1) transition strengths to the ground state, which could let measured E1 strengths constrain predictions.

What carries the argument

Equation (2) is the load-bearing object: the Schiff moment $S$ is a second-order sum over intermediate states $|\Psi_k\rangle$ with the same angular momentum as the ground state and opposite parity, $$S = \sum_{k} \frac{\langle\Psi_0|\hat{S}_z|\Psi_k\rangle\langle\Psi_k|\hat{V}_{PT}|\Psi_0\rangle}{E_0 - E_k} + \mathrm{c.c.},$$ with the nuclear-structure content packaged into the three structure factors $a_\alpha$. The generator coordinate method supplies the wave functions: superpositions of one-quasiparticle configurations with different quadrupole and octupole deformations, each projected onto good parity, neutron and proton number, and angular momentum. This machinery lets the authors turn on parity and particle-number projection, angular-momentum projection, and shape mixing one at a time, isolating which effects change $a_\alpha$ and which excited states carry the moment.

What would settle it

Computing the same structure factors with the giant-dipole resonances and multiquasiparticle states added to the basis, and finding the $^{129}$Xe or $^{199}$Hg values shift by more than the already-large spread among existing calculations, would show the basis is incomplete; conversely, a measured $B(E1)$ distribution that does not track the predicted state-by-state Schiff-moment contributions would call the claimed correlation into question.

Watch

Extended reading notes

Core claim

In second-order perturbation theory the nuclear Schiff moment is a sum over excited states of opposite parity but the same angular momentum as the ground state, weighting products of the Schiff-operator and $P,T$-odd-interaction matrix elements by energy denominators. The authors compute the three structure factors $a_\alpha$ that encode this nuclear-structure information in $^{129}$Xe, $^{199}$Hg, and $^{225}$Ra using generator-coordinate wave functions built from projected, deformed one-quasiparticle configurations. The central discovery is that beyond-mean-field correlations change the moments: in $^{225}$Ra the collective wave functions of the ground state and its lowest opposite-parity partner differ in the deformation plane, reducing their matrix elements and shrinking the octupole-enhanced $a_\alpha$ values by up to a factor of three compared with mean-field results, including those fitted to intrinsic octupole moments. In $^{129}$Xe and $^{199}$Hg many states up to about 20 MeV contribute comparably, and those contributions track the isovector $B(E1)$ strength connecting each state to the ground state. The final extrapolated structure factors are $a_0 = +0.15$, $a_1 = -1.9$, $a_2 = +1.3$ $e$ fm$^3$ for $^{225}$Ra, $-0.0052$, $+0.0134$, $-0.0097$ for $^{129}$Xe, and $-0.0163$, $+0.0327$, $-0.0375$ for $^{199}$Hg.

Load-bearing premise

The calculation assumes the generator-coordinate basis of axially deformed one-quasiparticle configurations includes all intermediate states that contribute meaningfully to the Schiff moment, even though the isovector giant-dipole resonance, much of the isoscalar giant-dipole resonance, and multiquasiparticle states are omitted.

Editorial extensions

If this is right

  • In $^{225}$Ra, the octupole enhancement of the Schiff moment is weaker than mean-field and octupole-constrained mean-field calculations indicate, with structure factors reduced by up to a factor of three.
  • The interpretation of $^{199}$Hg and $^{129}$Xe EDM limits depends more strongly than previously assumed on excited-state structure, because many intermediate states up to roughly 20 MeV contribute.
  • Measured $B(E1)$ transition strengths can serve as benchmarks to validate or rule out nuclear models that predict Schiff moments.
  • Models that fit intrinsic octupole moments still overestimate the $^{225}$Ra structure factors relative to the fully mixed calculation, so such fitting does not fully capture shape mixing.
  • Reliable predictions for $^{129}$Xe and $^{199}$Hg require an extended configuration space that includes the giant-dipole resonances and multiquasiparticle states.

Reading between the lines

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

  • If the $B(E1)$–Schiff-moment correlation survives when the giant-dipole resonance is added, measured E1 strength functions could act as a surrogate for Schiff moments, letting experiments constrain nuclear-structure uncertainty directly.
  • The shape-mixing suppression seen in $^{225}$Ra may operate in other octupole-soft and octupole-deformed nuclei, so systematic beyond-mean-field calculations could narrow the spread among Schiff-moment predictions across isotopic chains.
  • Because the simplified one-body form of $\hat{V}_{PT}$ is proportional to $\frac{d\rho}{dr}\boldsymbol{\sigma}\cdot\hat{\mathbf{r}}$, the omitted isovector giant-dipole resonance could itself participate in the correlation; extending the basis would show whether the correlation is a physical regularity or a truncation artifact.
  • The same framework could be applied to other EDM candidate nuclei and combined with a subspace-projected uncertainty quantification, giving the first statistical error bars on Schiff-moment structure factors.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. This manuscript presents the first multireference covariant density functional theory (MR-CDFT) calculations of nuclear Schiff moments for 129Xe, 199Hg, and 225Ra. The authors implement the generator coordinate method with parity, particle-number, and angular-momentum projection, and evaluate the second-order perturbation expression (Eq. (2)) for the structure factors a0, a1, and a2 associated with the pion-exchange P,T-odd interaction. After validating the low-energy spectra, magnetic moments, and B(E2) strengths, they report final values in Table I: 225Ra has much larger structure factors (+0.15, -1.9, +1.3 e fm3) dominated by the parity doublet, while 129Xe and 199Hg have smaller values (-0.0052, +0.0134, -0.0097 and -0.0163, +0.0327, -0.0375 e fm3) built from many intermediate states up to 20 MeV. The paper claims that beyond-mean-field correlations can either enhance or suppress Schiff moments, that shape mixing reduces the 225Ra enhancement by up to a factor of three, and that intermediate-state contributions correlate with B(E1) transition strengths.

Significance. If correct, this is a substantive advance: it is the first beyond-mean-field treatment of Schiff moments, it contains no Schiff-moment-specific fitted parameters (PC-PK1, the pairing interaction, and g_piNN are fixed from prior literature), and it validates the method on independent magnetic-dipole and electric-quadrupole data. The 225Ra result is made credible by the dominance of a single parity-doublet state whose energy and structure are reasonably reproduced. The B(E1) correlation, if confirmed by calculations with a larger intermediate-state space, would provide a practical bridge between E1 strength data and Schiff-moment predictions. The main caveat is that the 129Xe and 199Hg results are computed in a truncated GCM space; the paper acknowledges but does not quantify the omission of the isovector giant-dipole resonance, much of the isoscalar giant-dipole resonance, and multiquasiparticle states. The quantitative values for those two nuclei are therefore less secure than the 225Ra value.

major comments (4)
  1. [Formalism and Results; Eqs. (2), (6)] The central quantitative claim for 129Xe and 199Hg rests on Eq. (2), which sums over all intermediate states, but the basis in Eq. (6) is restricted to axially deformed one-quasiparticle configurations with quadrupole and octupole constraints. The text itself states that this basis 'may omit contributions from higher-lying states with different structure' and that such contributions 'could still be significant; Ref. [20] suggests that they are comparable to the contributions of low-lying states.' Because the Schiff operator in Eq. (3) is E1-weighted and V_PT is surface-peaked, the omitted isovector and much of the isoscalar giant-dipole resonance are not obviously negligible. Axial symmetry is also imposed, excluding triaxial fluctuations that can matter for soft nuclei. No estimate of the omitted contributions is provided, so the Table I entries for Xe and Hg are not demonstrated to be converged.
  2. [Results and discussion, Fig. 2] The running sums in Fig. 2(a,b) for 129Xe and 199Hg are still receiving contributions at the highest excitation energies shown (20 MeV), with no sign of saturation. This confirms that the reported structure factors depend on the truncation of the intermediate-state space. The authors should either extend the model space to include the giant resonances and multiquasiparticle states, or provide a quantitative bound on the omitted contributions before the Xe/Hg entries in Table I can be used as converged predictions.
  3. [Conclusions; Table I] The paper explicitly states that it has 'not attempted to quantify either the systematic or statistical uncertainties' of the predicted Schiff moments. Given that Table I is the central numerical output and Fig. 3 shows a large spread among models for Xe and Hg, the absence of any uncertainty estimate or sensitivity analysis (e.g., to the density functional, pairing interaction, or basis size) makes the comparison with other methods difficult to interpret. At minimum, a quantitative or semi-quantitative assessment of the expected size of the missing contributions should accompany the table.
  4. [Results and discussion, Fig. 2(d), and Conclusions] The claimed correlation between intermediate-state contributions to a_alpha and B(E1) is presented as a main result in the abstract and conclusions ('Our results reveal a strong correlation...'), but the body correctly hedges that the conclusion depends on whether the correlation 'survives the inclusion of the giant dipole resonance.' Because the GCM basis largely excludes the giant dipole resonance, the correlation is an intriguing but unverified observation. The abstract and conclusions should carry the same caveat, or the authors should test the correlation with a model that includes the GDR.
minor comments (4)
  1. [Table I and surrounding text] The table caption and the accompanying text both refer to 'Ra255' instead of '225Ra'; please correct this typo.
  2. [References] Refs. [26] and [51] are the same paper (Engel, Ramsey-Musolf, and van Kolck, 2013), as are Refs. [29] and [50] (Dobaczewski et al., 2018); please consolidate or cross-reference these duplicates.
  3. [Conclusions] The phrase 'first beyond-relativistic-mean-field studies' should read 'first beyond-mean-field studies' or 'first relativistic beyond-mean-field studies'; as written it conflates the relativistic framework with the beyond-mean-field aspect.
  4. [Results and discussion] Please specify whether the energy denominators in Eq. (2) use calculated or experimental excitation energies, particularly for the 225Ra parity-doublet state, where the predicted energy of 0.069 MeV differs from the experimental value of 0.055 MeV used in the figure.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the structure factors are genuine predictions from an externally benchmarked functional with no Schiff-moment data fitted, and the claimed B(E1) correlation is emergent from the model.

full rationale

The derivation chain is self-contained, and no step reduces to its own inputs by construction. Eq. (2) evaluates the Schiff moment in second-order perturbation theory with the standard Schiff operator (3) and the pion-exchange P,T-odd interaction (1); the only coupling constant, g_piNN = 12.9, is fixed externally by the Goldberger-Treiman relation (Ref. [33]), and the wave functions (5)-(6) are obtained from MR-CDFT with the PC-PK1 functional [40], a fixed pairing interaction, and no parameter adjusted to any Schiff-moment datum. The central quantities, the structure factors a_alpha in Table I, are outputs of the variational Hill-Wheeler-Griffin equations, not inputs. Validation is against independent external data: magnetic moments and B(E2) strengths are compared with NuDat experimental values (e.g., Ra225 101 W.u. vs 102(6) W.u.), so the model is externally falsifiable. The claimed B(E1) correlation is emergent: both the a_alpha running sums and the B(E1) strengths are computed from the same GCM states, and nothing in Eqs. (1)-(6) imposes the correlation. Self-citations to the MR-CDFT framework [31,35,39,41,47] and to Ref. [20] (de Jesus and Engel) share co-authors, but the framework is independently benchmarked, and Ref. [20] is used only to flag the acknowledged model-space incompleteness ('may omit contributions from higher-lying states... Ref. [20] suggests that they are comparable'), not to justify the central claim; no uniqueness theorem is imported. The explicit limitation statements in the Results and Conclusions (omission of the isovector and most of the isoscalar giant-dipole resonance and multiquasiparticle states; 'we have not attempted to quantify either the systematic or statistical uncertainties') are honest acknowledgments of accuracy and completeness limitations for Xe129 and Hg199, which is a correctness risk rather than a circularity. Accordingly the circularity score is 0.

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

The paper introduces no new fitted parameters for the target observables. The PC-PK1 functional and pairing parameters are calibrated in prior work; the oscillator basis size is a numerical truncation extrapolated to Nsh to infinity. The experimental energy of the Ra225 parity doublet is used in the figure, but no parameter is fitted to the Schiff moments themselves. The central claim rests on the standard second-order perturbation formula and on the completeness of the GCM basis, with the latter being the main unresolved assumption.

assumptions (6)
  • domain assumption The GCM ansatz of Eqs. (5)-(6), built from axially deformed one-quasiparticle configurations with constraints on Q20 and Q30, spans the intermediate states that contribute significantly to the Schiff moment sum in Eq. (2).
    Invoked throughout Results; the paper explicitly notes this omits the isovector giant-dipole resonance, much of the isoscalar giant-dipole resonance, and multiquasiparticle states, citing Ref. [20] for possible comparable contributions.
  • domain assumption Second-order perturbation theory in the P,T-odd pion-exchange interaction V_PT (Eq. 2), with unperturbed strong Hamiltonian states, is valid for computing Schiff moments.
    Standard in the field and used by all cited Schiff-moment calculations (Refs. [15,16,26]).
  • standard math The Schiff operator in Eq. (3), omitting the quadrupole correction and nucleon EDM contributions, is the relevant operator for diamagnetic atoms.
    Taken from Flambaum-Khriplovich-Sushkov and accepted within the atomic and nuclear EDM community.
  • domain assumption The relativistic energy density functional PC-PK1 plus BCS pairing with a delta interaction and smooth cutoff provides reliable mean-field configurations for Xe, Hg, and Ra isotopes.
    Borrowed from prior calibrations; validated here against magnetic moments and B(E2) values.
  • domain assumption Projection with 16 rotational and 7 gauge mesh points introduces spurious divergences and finite-step artifacts that have negligible impact on the results.
    Asserted in Numerical details with citations [42-46], but not demonstrated specifically for Schiff moments.
  • domain assumption Axial symmetry of the mean-field configurations, with K as a good quantum number, is sufficient for these nuclei.
    Stated in Numerical details; triaxial shapes and K-mixing are neglected without an explicit estimate of their effect on Schiff moments.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Effects of beyond-mean-field correlations on nuclear Schiff moments." pith.science (2026). https://pith.science/paper/52R7O2E5

@misc{pith2026250701369,
  author       = {Pith},
  title        = {Pith review of: Effects of beyond-mean-field correlations on nuclear Schiff moments},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/52R7O2E5}},
  note         = {Machine review of arXiv:2507.01369}
}
abstract

We compute the nuclear Schiff moments of the diamagnetic atoms $^{129}$Xe, $^{199}$Hg, and $^{225}$Ra in multireference covariant density functional theory. Beyond-mean-field correlations, arising from symmetry restoration and shape mixing, are incorporated via the generator coordinate method with projection onto states with well-defined parity, particle number, and angular momentum. Our results reveal a correlation between the contributions of nuclear intermediate states to Schiff moments and the electric dipole transition strengths from these states to the ground state. The new beyond-mean-field effects can either enhance or suppress the Schiff moments. In $^{225}$Ra, they do the latter, reducing the enhancement from octupole deformation somewhat.

Figures

Figures reproduced from arXiv: 2507.01369 by the authors.

Figure 1
Figure 1. FIG. 1. (Color online) The matrix elements of Schi [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Comparison of structure factors [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Discussion (0). Sign in to comment.

Forward citations

Cited by 5 Pith papers

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

  1. Limit on the nuclear Schiff moment of europium-153

    physics.atom-ph 2026-06 conditional novelty 7.0 of 10

    A crystal-based nuclear spin resonance measurement sets the first limit on the Schiff moment of 153Eu, |S|<1.7e-8 e fm^3, constraining TeV-scale T-violating new physics.

  2. Limit on the nuclear Schiff moment of europium-153

    physics.atom-ph 2026-06 unverdicted novelty 4.0 of 10

    Experimental upper limit |S(¹⁵³Eu)| < 1.7 × 10^{-8} e fm³ placed on the Schiff moment using nuclear spin resonances in polarized ¹⁵³Eu³⁺ ions in a Y₂SiO₅ crystal.

  3. The $^{229}$Th Isomer: Nuclear Structure, Clocks, and Tests of Fundamental Physics

    nucl-th 2026-06 accept novelty 3.0 of 10

    A review of 229Th isomer spectroscopy, clock platforms, nuclear-structure models, and fundamental-physics applications, with an additional mean-field check of the octupole minimum.

  4. The theory of electric dipole moments: the view from below

    hep-ph 2026-07 unverdicted novelty 2.0 of 10

    A bottom-up review organizing the theory of EDMs from quark-gluon CP violation via chiral perturbation theory and structure calculations, with emphasis on paramagnetic system sensitivity to hadronic sources.

  5. The $^{229}$Th Isomer: Nuclear Structure, Clocks, and Tests of Fundamental Physics

    nucl-th 2026-06 unverdicted novelty 1.0 of 10

    Review of the 229Th isomer covering spectroscopy, nuclear structure models, and applications to fundamental physics tests.

Reference graph

Works this paper leans on

52 extracted references · 25 canonical work pages · cited by 3 Pith papers

  1. [20]

    J. H. de Jesus and J. Engel, Phys. Rev. C 72, 045503 (2005), arXiv:nucl-th/0507031

    arXiv 2005

  2. [1]

    A. D. Sakharov, Pisma Zh. Eksp. Teor. Fiz. 5, 32 (1967)

    1967

  3. [2]

    Hocker and Z

    A. Hocker and Z. Ligeti, Ann. Rev. Nucl. Part. Sci. 56, 501 (2006), arXiv:hep-ph/0605217

    arXiv 2006

  4. [3]

    ’t Hooft, Phys

    G. ’t Hooft, Phys. Rev. Lett. 37, 8 (1976)

    1976

  5. [4]

    Chupp, P

    T. Chupp, P. Fierlinger, M. Ramsey-Musolf, and J. Singh, Rev. Mod. Phys. 91, 015001 (2019), arXiv:1710.02504 [physics.atom-ph]

    arXiv 2019

  6. [5]

    Graner, Y

    B. Graner, Y . Chen, E. G. Lindahl, and B. R. Heckel, Phys. Rev. Lett. 116, 161601 (2016), [Erratum: Phys.Rev.Lett. 119, 119901 (2017)], arXiv:1601.04339 [physics.atom-ph]

    arXiv 2016

  7. [6]

    Andreevet al

    V . Andreevet al. (ACME), Nature 562, 355 (2018)

    work page 2018

  8. [7]

    Abel et al., Phys

    C. Abel et al., Phys. Rev. Lett. 124, 081803 (2020)

    work page 2020

Show all 52 references
  1. [8]

    Alarcon et al., in Snowmass 2021 (2022) arXiv:2203.08103 [hep-ph]

    R. Alarcon et al., in Snowmass 2021 (2022) arXiv:2203.08103 [hep-ph]

    2022 arXiv

  2. [9]

    Yoshinaga, E

    N. Yoshinaga, E. Teruya, K. Higashiyama, and K. Yanase, JPS Conf. Proc. 23, 012034 (2018)

    2018

  3. [10]

    Sachdeva et al

    N. Sachdeva et al. , Phys. Rev. Lett. 123, 143003 (2019), arXiv:1902.02864 [physics.atom-ph]

    2019 arXiv

  4. [11]

    T. A. Zheng, Y . A. Yang, S. Z. Wang, J. T. Singh, Z. X. Xiong, T. Xia, and Z. T. Lu, Phys. Rev. Lett. 129, 083001 (2022), arXiv:2207.08140 [physics.atom-ph]

    2022 arXiv

  5. [12]

    B. C. Regan, E. D. Commins, C. J. Schmidt, and D. DeMille, Phys. Rev. Lett. 88, 071805 (2002)

    2002

  6. [13]

    R. H. Parker et al. , Phys. Rev. Lett. 114, 233002 (2015), arXiv:1504.07477 [nucl-ex]

    2015 arXiv

  7. [14]

    W. C. Haxton and E. M. Henley, Phys. Rev. Lett. 51, 1937 (1983)

    1983

  8. [15]

    V . V . Flambaum, I. B. Khriplovich, and O. P. Sushkov, Sov. Phys. JETP 60, 873 (1984)

    1984

  9. [16]

    V . V . Flambaum, I. B. Khriplovich, and O. P. Sushkov, Nucl. Phys. A 449, 750 (1986)

    1986

  10. [17]

    Spevak, N

    V . Spevak, N. Auerbach, and V . V . Flambaum, Phys. Rev. C56, 1357 (1997), arXiv:nucl-th/9612044

    1997 arXiv

  11. [18]

    V . F. Dmitriev and R. A. Sen’kov, Phys. Atom. Nucl. 66, 1940 (2003), arXiv:nucl-th/0304048

    2003 arXiv

  12. [19]

    V . F. Dmitriev, R. A. Sen’kov, and N. Auerbach, Phys. Rev. C 71, 035501 (2005), arXiv:nucl-th/0408065

    2005 arXiv

  13. [21]

    Dobaczewski and J

    J. Dobaczewski and J. Engel, Phys. Rev. Lett. 94, 232502 (2005), arXiv:nucl-th/0503057

    2005 arXiv

  14. [22]

    S. Ban, J. Dobaczewski, J. Engel, and A. Shukla, Phys. Rev. C 82, 015501 (2010), arXiv:1003.2598 [nucl-th]

    2010 arXiv

  15. [23]

    Teruya, N

    E. Teruya, N. Yoshinaga, K. Higashiyama, and K. Asahi, Phys. Rev. C 96, 015501 (2017)

    2017

  16. [24]

    Yoshinaga, K

    N. Yoshinaga, K. Yanase, and K. Higashiyama, J. Phys. Conf. Ser. 1643, 012006 (2020)

    2020

  17. [25]

    Yanase and N

    K. Yanase and N. Shimizu, Phys. Rev. C 102, 065502 (2020), arXiv:2006.15142 [nucl-th]

    2020 arXiv

  18. [27]

    J. S. M. Ginges and V . V . Flambaum, Phys. Rept. 397, 63 (2004), arXiv:physics/0309054

    2004 arXiv

  19. [28]

    Engel, (2025), 10.1146 /annurev-nucl-121423-101030, arXiv:2501.02744 [nucl-th]

    J. Engel, (2025), 10.1146 /annurev-nucl-121423-101030, arXiv:2501.02744 [nucl-th]

    2025 arXiv

  20. [30]

    Yanase, N

    K. Yanase, N. Shimizu, K. Higashiyama, and N. Yoshinaga, Phys. Lett. B 841, 137897 (2023), arXiv:2210.08498 [nucl-th]

    2023 arXiv

  21. [31]

    E. F. Zhou, X. Y . Wu, and J. M. Yao, Phys. Rev. C109, 034305 (2024), arXiv:2311.15305 [nucl-th]

    2024 arXiv

  22. [32]

    C. M. Maekawa, E. Mereghetti, J. de Vries, and U. van Kolck, Nucl. Phys. A 872, 117 (2011), arXiv:1106.6119 [nucl-th]

    2011 arXiv

  23. [33]

    M. L. Goldberger and S. B. Treiman, Phys. Rev. 110, 1178 (1958)

    1958

  24. [34]

    L. S. Song, J. M. Yao, P. Ring, and J. Meng, Phys. Rev. C 90, 054309 (2014)

    2014

  25. [35]

    J. M. Yao, E. F. Zhou, and Z. P. Li, Phys. Rev. C 92, 041304 (2015), arXiv:1507.03298 [nucl-th]

    2015 arXiv

  26. [36]

    Ring and P

    P. Ring and P. Schuck, The nuclear many-body problem (Springer-Verlag, New York, 1980)

    1980

  27. [37]

    D. L. Hill and J. A. Wheeler, Phys. Rev. 89, 1102 (1953)

    1953

  28. [38]

    J. J. Gri ffin and J. A. Wheeler, Phys. Rev.108, 311 (1957)

    1957

  29. [39]

    J. M. Yao and K. Hagino, Phys. Rev. C 94, 011303 (2016)

    2016

  30. [40]

    P. W. Zhao, Z. P. Li, J. M. Yao, and J. Meng, Phys. Rev. C 82, 054319 (2010)

    2010

  31. [41]

    J. M. Yao, J. Meng, P. Ring, and D. Vretenar, Phys. Rev. C 81, 044311 (2010), arXiv:0912.2650 [nucl-th]

    2010 arXiv

  32. [42]

    Anguiano, J

    M. Anguiano, J. L. Egido, and L. M. Robledo, Nucl. Phys. A 696, 467, arXiv:nucl-th/0105003

    arXiv

  33. [43]

    Tajima, H

    N. Tajima, H. Flocard, P. Bonche, J. Dobaczewski, and P. H. Heenen, Nucl. Phys. A 542, 355 (1992)

    1992

  34. [44]

    Dobaczewski, M

    J. Dobaczewski, M. V . Stoitsov, W. Nazarewicz, and P.-G. Reinhard, Phys. Rev. C 76, 054315 (2007)

    2007

  35. [45]

    Bender, T

    M. Bender, T. Duguet, and D. Lacroix, Phys. Rev. C79, 044319 (2009)

    2009

  36. [46]

    Duguet, M

    T. Duguet, M. Bender, K. Bennaceur, D. Lacroix, and T. Lesin- ski, Phys. Rev. C 79, 044320 (2009)

    2009

  37. [47]

    E. F. Zhou, X. Y . Wu, J. Xiang, J. M. Yao, and P. Ring, (2025), arXiv:2504.11244 [nucl-th]

    2025 arXiv

  38. [48]

    NuDat 2 Database,

    National Nuclear Data Center, “NuDat 2 Database,” (2020), https://www.nndc.bnl.gov/nudat2. 6

    2020

  39. [49]

    Engel, M

    J. Engel, M. Bender, J. Dobaczewski, J. H. De Jesus, and P. Olbratowski, Phys. Rev. C 68, 025501 (2003), arXiv:nucl- th/0304075

    2003

  40. [50]

    Dobaczewski, J

    J. Dobaczewski, J. Engel, M. Kortelainen, and P. Becker, Phys. Rev. Lett. 121, 232501 (2018), arXiv:1807.09581 [nucl-th]

    2018 arXiv

  41. [51]

    Engel, M

    J. Engel, M. J. Ramsey-Musolf, and U. van Kolck, Prog. Part. Nucl. Phys. 71, 21 (2013), arXiv:1303.2371 [nucl-th]

    2013 arXiv

  42. [52]

    Yoshinaga, K

    N. Yoshinaga, K. Higashiyama, R. Arai, and E. Teruya, Phys. Rev. C 87, 044332 (2013), [Erratum: Phys.Rev.C 89, 069902 (2014)]

    2013

  43. [53]

    de Vries, A

    J. de Vries, A. Gnech, and S. Shain, Phys. Rev. C103, L012501 (2021), arXiv:2007.04927 [hep-ph]

    2021 arXiv

  44. [54]

    Zhang, C

    X. Zhang, C. C. Wang, C. R. Ding, and J. M. Yao, (2024), arXiv:2408.00691 [nucl-th]

    2024 arXiv

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.