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Quark Phase Space Distributions in Nuclei

T0 review · 3 major / 3 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper claims that the fraction of baryons in a quarkyonic-like state tends to a fixed nuclear-matter limit for heavy nuclei, making low-momentum suppression testable.

desk verdict Careful Wigner machinery, but the fq>1 criterion is not a valid Pauli-violation indicator—a single nucleon already trips it, so the A-dependent plateau does not carry the physical claim. read the letter →

arxiv 2506.22670 v1 pith:MNNEYKAV submitted 2025-06-27 nucl-th

Alexis Nikolakopoulos , Gerald A. Miller This is my paper
classification nucl-th
keywords quarkyonicmatterWignerdistributionphase-spacedensitylow-momentumnucleonsuppressionlocalapproximationrelativisticmeanfieldHartree-Fock-Bogoliubovquasielasticelectronscattering
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 asks whether the quarkyonic suppression of low-momentum nucleons predicted for infinite nuclear matter can also be present in finite nuclei. The authors extend a previous nuclear-matter construction to nuclei by replacing Fermi-sea occupation numbers with the nuclear Wigner distribution, then convolve it with a quark momentum distribution to obtain a quark phase-space density $f_q$. Treating $f_q>1$ as a violation of the quark Pauli principle and integrating over the phase-space region where it occurs gives the fraction of baryons in a quarkyonic-like regime. The paper reports that for nuclei with mass number $A>50$ this fraction systematically approaches a fixed nuclear-matter limit, and that local-density and full mean-field calculations agree for large $A$. If true, the low-momentum suppression seen in extrapolated nuclear-matter electron-scattering data would be a genuine feature of heavy nuclei and testable by electron scattering.

What carries the argument

The central object is the nuclear Wigner quasi-probability distribution $W(r,p)$, obtained from the one-body density matrix, and its quark counterpart $W_q(r,k)$, obtained by convolution with the nucleon's quark momentum distribution $\phi$. Writing $f_q = w_q/2$, the condition $f_q>1$ marks phase-space regions where the quark Pauli bound would be exceeded; the probability $P_{f_q>1}$ is the fraction of baryons in that region. Because the convolution probes the one-body density matrix only at separations $s \lesssim R_N$ (the nucleon radius), the local-density approximation reproduces the full mean-field result for heavy nuclei, which is what makes the $A$-dependence systematic.

What would settle it

A measurement of the missing-momentum distribution in $(e,e'p)$ knockout from $^{208}$Pb that resolves deep-lying $s$-states: standard mean-field momentum distributions rise smoothly toward $p_m=0$, whereas this model predicts suppression below roughly 20-30 percent of the Fermi momentum, so seeing no dip would rule out the suppression.

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Extended reading notes

Core claim

The central claim is that the fraction of baryons occupying quark phase space beyond the Pauli bound, $P_{f_q>1}$, computed from the convolution of the nuclear Wigner distribution with a simple quark momentum distribution, systematically tends to a constant as $A$ increases beyond roughly 50. This holds both in the local-density approximation and in full independent-particle calculations with Hartree-Fock-Bogoliubov and relativistic mean-field wavefunctions. The associated ratio $k_1/k_F$, the size of the 'hole' in the nucleon momentum distribution in the quarkyonic model, saturates at 20-30 percent for $A>100$. The paper therefore concludes that low-momentum nucleon suppression is plausible in heavy finite nuclei and can be probed by inclusive $(e,e')$ and exclusive $(e,e'p)$ measurements.

Load-bearing premise

The argument rests on interpreting the angle-averaged convolved Wigner distribution $w_q/2$ as a genuine quark occupation number, so that the Pauli bound $f_q \le 1$ applies; the paper explicitly notes this interpretation can be questioned, and if it fails the probability $P_{f_q>1}$ loses its physical meaning.

Editorial extensions

If this is right

  • For $A>50$, the fraction of baryons in the quarkyonic-like region plateaus, so the effect should be visible in medium and heavy nuclei, not only in extrapolated nuclear matter.
  • The size of the low-momentum hole $k_1/k_F$ is 20-30 percent, comparable to the nuclear-matter prediction, so electron-scattering data used to extrapolate to nuclear matter should already contain the suppression.
  • Because local-density and full Wigner-distribution calculations agree for heavy nuclei, predictions for nuclei like $^{208}$Pb are not sensitive to the choice of mean-field functional.
  • The paper identifies concrete electron-scattering tests, including inclusive $(e,e')$ near $x \approx 1$ and exclusive $(e,e'p)$ knockout of deep-lying $s$-states, that could rule the model out.

Reading between the lines

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

  • The Wigner distribution's necessary negativity means $f_q$ can at best be a quasi-occupation number; checking whether the plateau survives with a positive-definite phase-space representation, such as a Husimi distribution, is a natural extension the paper does not perform.
  • If the suppression is real, it would modify the nucleon spectral functions used in neutrino-nucleus scattering generators, since those typically assume standard independent-nucleon momentum distributions.
  • The plateau for $A>50$ implies that the most decisive experiments could be run on $^{40}$Ca or $^{48}$Ca, where the predicted effect is already near saturation but the nuclear theory is more tractable than for $^{208}$Pb.
  • A direct confrontation with data would compare the predicted $f_q(r,k)$ with momentum distributions extracted from $(e,e'p)$ on $^{40}$Ca and $^{208}$Pb at low missing momentum, where the model predicts a depletion that standard shell-model fits would not produce.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 3 minor

Summary. The paper extends the IdylliQ quarkyonic-matter construction of Koch, McLerran, Miller, and Vovchenko to finite nuclei. The authors replace nuclear-matter occupation numbers by the nuclear Wigner distribution, convolve it with a model quark momentum distribution, and define fq(r,p)=wq(r,p)/2 as a quark phase-space occupation. They then define Pfq>1, the fraction of baryons in regions where fq>1, as the fraction of baryons in a quarkyonic-like regime. Using LDA, RMF, and HFB calculations for nuclei from 12C to 238U, they find that Pfq>1 tends to a constant for A>50, interpret this as evidence that low-momentum suppression is plausible in heavy nuclei, and propose electron-scattering tests. The technical comparison between LDA and full Wigner results is carefully done, but the physical interpretation of fq>1 is the central weak point.

Significance. If the fq>1 criterion were a valid measure of quark Pauli oversaturation, the paper would provide a useful bridge between the IdylliQ model of infinite nuclear matter and finite nuclei, with concrete experimental consequences. The Wigner-distribution machinery, the systematic LDA-versus-full comparison, and the proposed (e,e'p) and pion-production tests are valuable and could be reused in other contexts. However, the central plausibility claim rests entirely on identifying wq/2 as a quark occupation number, an identification that the paper itself questions and that a single-nucleon counterexample shows to be invalid. As a result, the main physical conclusion is not currently supported; the paper is better viewed as a model-internal diagnostic study unless the occupation-number interpretation can be justified from a microscopic many-quark state.

major comments (3)
  1. [Sec. III.B and III.C, Eq. (27)] The identification fq(r,p)=wq(r,p)/2 as a quark occupation number is the load-bearing step, but it is assumed rather than derived. A single 1s harmonic-oscillator nucleon, whose three quarks are in a color-antisymmetric wave function and therefore cannot violate the quark Pauli principle, already yields fq(0,0)>1 for the model of Eq. (22) with typical parameters: using W(0,0)=8, saturating the bound of Eq. (A13), and Λ=240 MeV with b≈1 fm gives fq(0,0) of order unity or larger. Thus the criterion fq>1 does not reliably signal Pauli oversaturation, and Pfq>1 in Eq. (27) has no established physical meaning. The manuscript explicitly acknowledges this in Sec. III.B ('the reader might question whether fq can be interpreted as the occupation probability of quarks in a Fermi gas type system'), but the subsequent analysis proceeds without resolving the issue. Since the A-dependence result of Sec. III.C is built entirely on this quantity, the central plausibility conclusion is not supported.
  2. [Sec. III.A, Eq. (25)] The normalization of fq with respect to color degeneracy is unclear and affects the critical-density condition. If fq is summed over color, the Pauli bound is Nc, not 1; if fq is per color, the factor Nc inside Eq. (25) requires justification. As written, the right-hand side of Eq. (25) tends to Nc^3 for kF≫ΛNc, which is difficult to reconcile with any ordinary occupation-number interpretation. This ambiguity propagates into the critical densities shown in Fig. 2 and the boundary of the fq>1 region used throughout the paper, so it should be clarified before the numerical results can be assessed.
  3. [Sec. III.C, Fig. 6] The linear growth of Nq with A, and hence the plateau in Pfq>1, is expected from the geometry of a constant-density sphere, as the authors themselves note ('In a simplified model for the nucleus of a sphere with constant density this relation would be exact'). The plateau therefore does not by itself discriminate quarkyonic physics from any model with a fixed interior phase-space density. Its physical content would be restored only if fq had a valid occupation-number interpretation, which is exactly the point raised in the previous comments. The agreement between LDA and full Wigner results is a useful technical finding, but it does not strengthen the physical interpretation of the plateau.
minor comments (3)
  1. [Fig. 6 caption and Sec. III.C] The right-hand panel of Fig. 6 is labelled 'k1/k2' while the text refers to 'k1/kF' and 'khole/kF'; these notations should be unified for clarity.
  2. [Eq. (24)] The sentence 'denote the equivalent linear combinations of other quantities with the same subscripts' is vague; it would be clearer to write explicitly how the up/down quark Wigner distributions are constructed from the proton and neutron Wigner distributions.
  3. [Sec. V, final paragraph] The statement that 'a code to compute the Wigner distributions is available upon request' would be more reproducible if the code were deposited in a permanent public repository with versioning and usage documentation.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular reduction: the f_q>1 quarkyonic fraction is a conditional model measure, and the large-A plateau is a computed consequence of nuclear saturation plus the model's critical density, not a fitted parameter or an equation that reduces to its input.

full rationale

The paper is a conditional extension of a phenomenological model, not a first-principles derivation, and no load-bearing step reduces to its own input by construction. The nuclear Wigner distributions are computed from independent mean-field inputs (SLY4 HFB and NL-SH RMF wavefunctions), and the quark phase-space density W_q(r,k) is obtained by the explicit convolution of Eq. (18)/(20) with the quark momentum distribution of Refs. [19,20]. The central quantity P_{f_q>1} in Eq. (27) is not fitted to any data: it integrates the computed w_q over the region where w_q/2>1, and its A-dependence is evaluated numerically for both the full one-body density matrix and the LDA. The authors state that the linear large-A trend is expected in a constant-density sphere ('The linear relation is expected'), so the plateau is presented as a consistency check rather than as an independent empirical prediction. The main fragility is the identification of w_q/2 as a quark occupation number, which is imported from the self-cited model of Ref. [21] and explicitly questioned by the authors in Sec. III.B ('the reader might question whether f_q can be interpreted as the occupation probability of quarks in a Fermi gas type system...'). That is an interpretive assumption, and a single-nucleon Wigner distribution would indeed saturate the bound of Eq. (A13) and can yield f_q>1 after convolution, undermining the Pauli-violation reading; but this is a validity or correctness concern, not a circular derivation. Self-citation supplies the model being extended, not a uniqueness theorem or a no-alternatives argument, and the numerical quantities do not collapse into their definitions. Accordingly, the circularity score is low; the definitional caveat prevents a clean 0 but does not constitute a circular step.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The analysis rests on a parameterized quark momentum distribution (Lambda) and on the interpretive assumption that fq>1 marks quarkyonic behavior. No new particles or forces are introduced.

free parameters (1)
  • Lambda (quark momentum distribution width) = 200-280 MeV (varied)
    Controls the exponential quark momentum distribution in Eq. (22) and the size of the fq>1 region; not fitted to data in this paper, inherited from the IdylliQ framework.
assumptions (5)
  • ad hoc to paper The quark momentum distribution in a nucleon is the exponential model of Eq. (22).
    Taken from Refs. [19,20]; not derived from QCD. Determines the convolution kernel between quark and nucleon Wigner distributions.
  • domain assumption The nucleus is described by an independent-particle Slater determinant (HFB or RMF).
    Used to construct the one-body density matrix and Wigner distribution; neglects short-range correlations that affect nuclear momentum distributions.
  • ad hoc to paper The convolved distribution wq(r,k)/2 is interpreted as a quark occupation number, with fq>1 indicating a violation of the quark Pauli principle and the presence of quarkyonic degrees of freedom.
    Central interpretive step; the paper itself notes the reader might question this interpretation (Sec. III.B).
  • domain assumption The local density approximation, with local Fermi momentum kF(r), is accurate for computing quark phase-space distributions.
    Relies on the one-body density matrix matching the nuclear matter form at small separations s; checked numerically but approximate for light nuclei.
  • standard math Standard properties of Wigner distributions, including their definition and possible negativity.
    Background used throughout; Eq. (3) defines the Wigner distribution.

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Pith. "Pith review of Quark Phase Space Distributions in Nuclei." pith.science (2026). https://pith.science/paper/MNNEYKAV

@misc{pith2026250622670,
  author       = {Pith},
  title        = {Pith review of: Quark Phase Space Distributions in Nuclei},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MNNEYKAV}},
  note         = {Machine review of arXiv:2506.22670}
}
abstract

In [PRC 110, 025201], the authors construct a model for nuclear matter which features a quarkyonic phase. A main feature in this model is that the nucleon occupation is strongly reduced at small momenta. Somewhat surprisingly, this result is supported by data for electron scattering from nuclear matter, where a reduction of the cross section consistent with suppression of nucleons with small momenta is seen. Since nuclear matter data are obtained by extrapolation of electron scattering data on increasingly heavier systems, this feature should manifest at least to some degree in heavy nuclei. To check if this is plausible we extend the approach of [PRC 110, 025201] to finite nuclei by considering the nuclear Wigner distribution. We use non-relativistic and relativistic independent particle models to determine the nuclear Wigner distribution, in addition to the local-density approximation (LDA). Phase-space distributions of quarks are obtained as a convolution of the Wigner distribution with a quark momentum distribution. We highlight some properties of the Wigner distribution in spherical systems, which can spoil the interpretation of the quark phase-space distribution as occupation numbers in the Fermi sea. On the other hand, we show that large systems behave essentially like infinite nuclear matter in their interior, and that LDA and full results are quantitatively similar for large A. We then compute the fraction of baryons that would be in a quarkyonic phase in the same sense as in [PRC 110, 025201] for a set of nuclei with mass $12 \leq A \leq 238$. We find that this fraction systematically tends to a constant at large $A$. It is hence plausible that the suppression seen in the nuclear matter data is a genuine feature, present in large finite nuclei. This result is counter-intuitive and we discuss possible electron scattering measurements that could rule out this model.

Figures

Figures reproduced from arXiv: 2506.22670 by the authors.

Figure 1
Figure 1. FIG. 1. Angle averaged Wigner distribution [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. The value of [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Region of phase space where [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Top panels: angle-averaged one-body density matri [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Quark occupation numbers [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Number of baryons for which [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]

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Cited by 1 Pith paper

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

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Reference graph

Works this paper leans on

47 extracted references · 28 canonical work pages · cited by 1 Pith paper

  1. [1]

    Hagen, T

    G. Hagen, T. Papenbrock, M. Hjorth-Jensen, and D. J. Dean, Rept. Prog. Phys. 77, 096302 (2014), arXiv:1312.7872 [nucl-th]

    arXiv 2014

  2. [2]

    B. R. Barrett, P. Navratil, and J. P. Vary, Prog. Part. Nucl. Phys. 69, 131 (2013)

    work page 2013

  3. [3]

    Hergert, S

    H. Hergert, S. K. Bogner, T. D. Morris, A. Schwenk, and K. Tsukiyama, Phys. Rept. 621, 165 (2016), arXiv:1512.06956 [nucl-th]

    arXiv 2016

  4. [4]

    Geesaman, K

    D. Geesaman, K. Saito, and A. Thomas, Ann. Rev. Nucl. and Part. Sci. 45, 337 (1995)

    work page 1995

  5. [5]

    P. R. Norton, Rept. Prog. Phys. 66, 1253 (2003)

    work page 2003

  6. [6]

    O. Hen, D. W. Higinbotham, G. A. Miller, E. Piasetzky, and L. B. Weinstein, Int. J. Mod. Phys. E 22, 1330017 (2013), arXiv:1304.2813 [nucl-th]

    arXiv 2013

  7. [7]

    O. Hen, G. A. Miller, E. Piasetzky, and L. B. Weinstein, Rev. Mod. Phys. 89, 045002 (2017), arXiv:1611.09748 [nucl-ex]

    arXiv 2017

  8. [8]

    McLerran and R

    L. McLerran and R. D. Pisarski, Nucl. Phys. A 796, 83 (2007), arXiv:0706.2191 [hep-ph]

    arXiv 2007

Show all 47 references
  1. [9]

    McLerran and S

    L. McLerran and S. Reddy, Phys. Rev. Lett. 122, 122701 (2019), arXiv:1811.12503 [nucl-th]

    2019 arXiv

  2. [10]

    Fujimoto, K

    Y. Fujimoto, K. Fukushima, and K. Murase, Phys. Rev. D 101, 054016 (2020), arXiv:1903.03400 [nucl-th]

    2020 arXiv

  3. [11]

    H. Tan, J. Noronha-Hostler, and N. Yunes, Phys. Rev. Lett. 125, 261104 (2020), arXiv:2006.16296 [astro- ph.HE]

    2020 arXiv

  4. [12]

    K. S. Jeong, L. McLerran, and S. Sen, Phys. Rev. C 101, 035201 (2020), arXiv:1908.04799 [nucl-th]

    2020 arXiv

  5. [13]

    Kojo, Phys

    T. Kojo, Phys. Rev. D 104, 074005 (2021), arXiv:2106.06687 [nucl-th]

    2021 arXiv

  6. [14]

    Kovensky and A

    N. Kovensky and A. Schmitt, JHEP 09, 112 (2020), arXiv:2006.13739 [hep-th]

    2020 arXiv

  7. [15]

    Cao and J

    G. Cao and J. Liao, JHEP 10, 168 (2020), arXiv:2007.02028 [nucl-th]

    2020 arXiv

  8. [16]

    D. C. Duarte, S. Hernandez-Ortiz, and K. S. Jeong, Phys. Rev. C 102, 025203 (2020), arXiv:2003.02362 [nucl-th]

    2020 arXiv

  9. [17]

    Sen and N

    S. Sen and N. C. Warrington, Nucl. Phys. A1006, 122059 (2021), arXiv:2002.11133 [nucl-th]. 10

    2021 arXiv

  10. [18]

    R. V. Poberezhnyuk, H. Stoecker, and V. Vovchenko, Phys. Rev. C 108, 045202 (2023), arXiv:2307.13532 [nucl-th]

    2023 arXiv

  11. [19]

    Koch and V

    V. Koch and V. Vovchenko, Phys. Lett. B 841, 137942 (2023), arXiv:2211.14674 [nucl-th]

    2023 arXiv

  12. [20]

    Fujimoto, T

    Y. Fujimoto, T. Kojo, and L. D. McLerran, Phys. Rev. Lett. 132, 112701 (2024), arXiv:2306.04304 [nucl-th]

    2024 arXiv

  13. [21]

    V. Koch, L. McLerran, G. A. Miller, and V. Vovchenko, Phys. Rev. C 110, 025201 (2024), arXiv:2403.15375 [nucl-th]

    2024 arXiv

  14. [22]

    D. B. Day et al., Phys. Rev. C 40, 1011 (1989)

    1989

  15. [23]

    Wigner, Phys

    E. Wigner, Phys. Rev. 40, 749 (1932)

    1932

  16. [24]

    NIST Digital Library of Mathematical Functions (10.22.63),

    F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, B. V. Saunders, H. S. Cohl, and M. A. Mc- Clain, “NIST Digital Library of Mathematical Functions (10.22.63),” https://dlmf.nist.gov/10.22.63, Release 1.2.4 of 2025-03-15 ()

  17. [25]

    Bennaceur and J

    K. Bennaceur and J. Dobaczewski, Comput. Phys. Com- mun. 168, 96 (2005), arXiv:nucl-th/0501002

    2005 arXiv

  18. [26]

    P¨ oschl, D

    W. P¨ oschl, D. Vretenar, A. Rummel, and P. Ring, Com- puter Physics Communications 101, 75 (1997)

    1997

  19. [27]

    The relativistic impulse approximation,

    C. J. Horowitz, D. P. Murdock, and B. D. Serot, “The relativistic impulse approximation,” in Computational Nuclear Physics 1: Nuclear Structure, edited by K. Lan- ganke, J. A. Maruhn, and S. E. Koonin (Springer Berlin Heidelberg, Berlin, Heidelberg, 1991) pp. 129–151

    1991

  20. [28]

    Sharma, M

    M. Sharma, M. Nagarajan, and P. Ring, Physics Letters B 312, 377 (1993)

    1993

  21. [29]

    Chabanat, P

    E. Chabanat, P. Bonche, P. Haensel, J. Meyer, and R. Schaeffer, Nuclear Physics A 635, 231 (1998)

    1998

  22. [30]

    Cosyn and J

    W. Cosyn and J. Ryckebusch, Phys. Lett. B 820, 136526 (2021), arXiv:2106.01249 [nucl-th]

    2021 arXiv

  23. [31]

    J. W. Negele and D. Vautherin, Phys. Rev. C 5, 1472 (1972)

    1972

  24. [32]

    R. R. Whitney, I. Sick, J. R. Ficenec, R. D. Kephart, and W. P. Trower, Phys. Rev. C9, 2230 (1974)

    1974

  25. [33]

    Zghiche et al., Nucl

    A. Zghiche et al., Nucl. Phys. A572, 513 (1994)

    1994

  26. [34]

    Boffi, C

    S. Boffi, C. Giusti, and F. D. Pacati, Phys. Rep. 226, 1 (1993)

    1993

  27. [35]

    Lapikas, G

    L. Lapikas, G. van der Steenhoven, L. Frankfurt, M. Strikman, and M. Zhalov, Phys. Rev. C 61, 064325 (2000), arXiv:nucl-ex/9905009

    2000 arXiv

  28. [36]

    J. J. Kelly, Phys. Rev. C 71, 064610 (2005)

    2005

  29. [37]

    Lapikas, Nucl

    L. Lapikas, Nucl. Phys. A 553, 297c (1993)

    1993

  30. [38]

    Kolar et al

    T. Kolar et al. (A1), Phys. Lett. B 847, 138309 (2023), arXiv:2306.05565 [nucl-ex]

    2023 arXiv

  31. [39]

    Kolar et al

    T. Kolar et al. (A1), Phys. Rev. C 110, L061302 (2024), arXiv:2311.01935 [nucl-ex]

    2024 arXiv

  32. [40]

    X. Li, L. E. Wright, and C. Bennhold, Phys. Rev. C 48, 816 (1993)

    1993

  33. [41]

    Glavanakov, V

    I. Glavanakov, V. Stibunov, et al., Sov. J. Nucl. Phys 80, 465 (1979)

    1979

  34. [42]

    Alvarez-Ruso, M

    L. Alvarez-Ruso, M. Sajjad Athar, M. Barbaro, D. Cher- dack, M. Christy, P. Coloma, T. Donnelly, S. Dytman, A. de Gouvˆ ea, R. Hill, P. Huber, N. Jachowicz, T. Katori, A. Kronfeld, K. Mahn, M. Martini, J. Morf ´ ın, J. Nieves, G. Perdue, R. Petti, D. Richards, F. S´ anchez, T....

    2018

  35. [43]

    Bechler, 26, 6039 (1993)

    A. Bechler, 26, 6039 (1993)

    1993

  36. [44]

    NIST Digital Library of Mathematical Functions (10.22.26),

    F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, B. V. Saunders, H. S. Cohl, and M. A. Mc- Clain, “NIST Digital Library of Mathematical Functions (10.22.26),” https://dlmf.nist.gov/10.22.26, Release 1.2.4 of 2025-03-15 ()

  37. [45]

    Multi-dimensional adaptive integration in C: The Cubature package,

    S. G. Johnson, “Multi-dimensional adaptive integration in C: The Cubature package,” https://github.com/ stevengj/cubature (2005). Appendix A: Wigner distribution in the Relativistic Mean Field In the relativistic mean field, the nuclear ground state is described as a Slater de...

    2005

  38. [46]

    Properties of the Wigner distribution The distribution is normalized so that Z dp (2π)3 W (r, p) = X κ 2Jκ + 1 4π g2 κ(r) + f 2 κ(r) = ρ(r, r) (A9) is the baryon density. The Wigner distribution has the same form for r and p-space wavefunctions Wκ(r, p) = Z dseir·sρκ(p − s/2, ...

  39. [47]

    In this case, one can write the wavefunction as the two com- ponent wavefunction ψmj n,κ = un,κ(r)Φmj κ (Ωr)

    Non relativistic wavefunctions The discussion above remains practically unchanged when non-relativistic wavefunctions are considered. In this case, one can write the wavefunction as the two com- ponent wavefunction ψmj n,κ = un,κ(r)Φmj κ (Ωr) . (A18) All results are then the s...

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