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arxiv: 1907.10986 · v1 · pith:WXXNQK4Unew · submitted 2019-07-25 · ⚛️ nucl-th

Neutron-proton pairing in Nuclear Matter

Xiao-Hua Fan , Xin-le Shang , Jian-min Dong , Wei Zuo This is my paper

Pith reviewed 2026-05-24 16:02 UTC · model grok-4.3

classification ⚛️ nucl-th
keywords neutron-proton pairingnuclear matterself-energy effectpairing gapeffective pairing forceBrueckner-Hartree-Fockthree-body force
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The pith

Self-energy effects strongly modify the neutron-proton pairing gap, and the results calibrate an effective force for nuclei.

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

The paper investigates the impact of self-energy on the neutron-proton pairing gap in nuclear matter using an extended Brueckner-Hartree-Fock approach with BCS theory. It finds that second-order self-energy contributions reduce the gap substantially while the renormalization term increases it notably. The three-body force has negligible influence on the gap. These microscopic results are then used to determine parameters for a density-dependent zero-range pairing interaction suitable for finite nuclei calculations.

Core claim

Within the extended BHF approach combined with BCS theory, the self-energy up to the second-order contribution reduces strongly the effective energy gap, while the renormalization term enhances it significantly. The effect of the three-body force on the np pairing gap is negligible. An effective density-dependent zero-range pairing force is established with parameters calibrated to the microscopically calculated energy gap to connect with np pairing in finite nuclei.

What carries the argument

Extended Brueckner-Hartree-Fock approach combined with BCS theory applied to self-energy contributions in neutron-proton pairing.

If this is right

  • The effective np pairing gap is reduced by second-order self-energy effects.
  • The renormalization term provides significant enhancement to the gap.
  • Three-body forces have negligible impact on the gap.
  • The microscopic gap allows calibration of a simple effective pairing force for finite nuclei.

Where Pith is reading between the lines

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

  • This calibration could improve modeling of pairing correlations in finite nuclei by supplying a practical contact force.
  • The negligible three-body force effect indicates that two-body interactions dominate the gap at the densities considered.
  • Similar self-energy analysis could be applied to other pairing channels such as neutron-neutron or proton-proton.

Load-bearing premise

The extended BHF approach combined with BCS theory accurately captures the self-energy contributions to the np pairing gap when using the chosen nucleon-nucleon interaction.

What would settle it

An independent many-body calculation of the neutron-proton pairing gap at nuclear saturation density that shows a value differing substantially from the self-energy corrected result.

Figures

Figures reproduced from arXiv: 1907.10986 by Jian-min Dong, Wei Zuo, Xiao-Hua Fan, Xin-le Shang.

Figure 1
Figure 1. Figure 1: FIG. 1: Hole-line expansion of the self-energy. [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: (Color online). The real and imaginary parts (upper p [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: (Color online). Neutron-proton effective energy gap i [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: (Color online). The np pairing gap using the effective d [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
read the original abstract

The self-energy effect on the neutron-proton (np) pairing gap is investigated up to the third order within the framework of the extend Bruecker-Hartree-Fock (BHF) approach combined with the BCS theory. The self-energy up to the second-order contribution turns out to reduce strongly the effective energy gap, while the \emph{renormalization} term enhances it significantly. In addition, the effect of the three-body force on the np pairing gap is shown to be negligible. To connect the present results with the np pairing in finite nuclei, an effective density-dependent zero-range pairing force is established with the parameters calibrated to the microscopically calculated energy gap.

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

2 major / 1 minor

Summary. The paper examines the effects of self-energy contributions up to third order on the neutron-proton pairing gap in nuclear matter using the extended Brueckner-Hartree-Fock approach combined with BCS theory. It finds that the second-order self-energy reduces the gap significantly, the renormalization term enhances it, and three-body forces have a negligible effect. An effective density-dependent zero-range pairing force is then calibrated to these microscopic results for application to finite nuclei.

Significance. If the numerical trends hold, the order-by-order decomposition of self-energy effects on the np gap provides useful insight into many-body contributions in nuclear matter. The negligible impact of three-body forces is a clear result worth noting. The calibrated effective force could serve as a practical tool for finite-nucleus calculations, though its transferability requires further scrutiny.

major comments (2)
  1. [Abstract] Abstract: The effective density-dependent zero-range pairing force parameters are calibrated directly to the microscopically computed gap obtained from the same extended BHF+BCS calculation. This makes the force for finite nuclei dependent on the paper's own output rather than an independent benchmark, which is load-bearing for the central claim of connecting the nuclear-matter results to np pairing in finite nuclei.
  2. [Abstract] Abstract: No error estimates on the gap values, no details on the specific nucleon-nucleon interaction employed, and no verification that the order-by-order self-energy separation is free of double-counting are provided. These omissions undermine assessment of the reported trends (second-order reduction and renormalization enhancement).
minor comments (1)
  1. [Abstract] The abstract contains the typographical error 'extend Bruecker-Hartree-Fock' (should be 'extended Brueckner-Hartree-Fock').

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major comment point by point below, indicating planned revisions where appropriate.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The effective density-dependent zero-range pairing force parameters are calibrated directly to the microscopically computed gap obtained from the same extended BHF+BCS calculation. This makes the force for finite nuclei dependent on the paper's own output rather than an independent benchmark, which is load-bearing for the central claim of connecting the nuclear-matter results to np pairing in finite nuclei.

    Authors: We agree that the effective pairing force is calibrated directly to the gaps obtained from our extended BHF+BCS calculations. This is intentional, as the goal is to transfer the microscopic nuclear-matter results into a practical form for finite-nucleus applications. However, we acknowledge that this calibration is not an independent benchmark. In the revised manuscript we will explicitly state in the abstract and conclusions that the force parameters are fitted to the computed gaps from the present calculations and will discuss the implications for transferability. revision: yes

  2. Referee: [Abstract] Abstract: No error estimates on the gap values, no details on the specific nucleon-nucleon interaction employed, and no verification that the order-by-order self-energy separation is free of double-counting are provided. These omissions undermine assessment of the reported trends (second-order reduction and renormalization enhancement).

    Authors: We will add the specific nucleon-nucleon interaction (Argonne V18) to the revised abstract and methods section. The order-by-order separation in the extended BHF framework is constructed without double-counting: the second-order self-energy is the explicit perturbative correction while the renormalization term is the quasiparticle strength factor Z applied to the single-particle energies; these are distinct and non-overlapping contributions by definition of the approach. We will insert a brief clarifying sentence on this point. Quantitative error estimates on the gap values are not provided because the calculations are deterministic within the chosen truncation; a discussion of numerical sensitivity can be added, but full uncertainty quantification from many-body truncation lies beyond the present scope. revision: partial

Circularity Check

1 steps flagged

Effective pairing force parameters calibrated directly to paper's own microscopic gap calculation

specific steps
  1. fitted input called prediction [Abstract]
    "To connect the present results with the np pairing in finite nuclei, an effective density-dependent zero-range pairing force is established with the parameters calibrated to the microscopically calculated energy gap."

    The parameters of the effective force are fitted to the energy gap value obtained from the paper's own extended BHF+BCS calculation, so the claimed connection to finite nuclei is constructed by direct calibration to the paper's microscopic result rather than an external or independent constraint.

full rationale

The paper computes the np pairing gap in nuclear matter using extended BHF+BCS, then directly calibrates an effective density-dependent zero-range force to that computed gap for use in finite nuclei. This calibration step makes the effective force a fitted representation of the paper's own output rather than an independent benchmark or derivation. The abstract explicitly states the calibration, matching the fitted_input_called_prediction pattern with no additional self-citation or definitional circularity in the provided text.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The work rests on the validity of the extended BHF approximation and BCS mean-field treatment for pairing, plus the assumption that a zero-range density-dependent force can faithfully represent the microscopic gap; no new entities are introduced.

free parameters (1)
  • effective pairing force parameters
    Strength and density dependence of the zero-range force are fitted to match the computed microscopic np gap.
axioms (2)
  • domain assumption Extended Brueckner-Hartree-Fock plus BCS theory provides a controlled expansion for self-energy effects on the pairing gap
    Invoked throughout the abstract as the framework for separating second-order self-energy, renormalization, and third-order terms.
  • domain assumption Three-body force contributions can be added perturbatively without altering the gap significantly
    Stated as a result but relies on the specific three-body force model chosen in the BHF calculation.

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Reference graph

Works this paper leans on

47 extracted references · 47 canonical work pages

  1. [1]

    which is also true for the exact gap of equation (9). The presen ce of the quasiparticle strength, which is less than unitary in a small region around the Ferm i surface where the Cooper pairs are mainly formed, reduces the pairing gap. The gap equation should be solved self-consistently with the density constraint since the pairing could modify the chemi...

    work page

  2. [2]

    As mentioned in the introductio n, the imaginary part of Σ( k, ω ) goes to zero at the Fermi energy

    75kF is exhibited in the upper panel of Fig.2. As mentioned in the introductio n, the imaginary part of Σ( k, ω ) goes to zero at the Fermi energy. This is true for the momentum k = kF as well, which implies the quasiparticle strength approximation is reliable near kF . However, ImΣ becomes sizable compared to the real part of the se lf-energy at the s.p....

    work page

  3. [3]

    The present obtained effective gap in the ex tended BHF approach turns out to be slightly larger than that in the T matrix approximation in the density region of ρ > 0

    is shown in the inset. The present obtained effective gap in the ex tended BHF approach turns out to be slightly larger than that in the T matrix approximation in the density region of ρ > 0. 045f m− 3, while it becomes a bit smaller than that in the T matrix approximation 9 /s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48 /s49...

    work page

  4. [4]

    Bohr, B.R

    A. Bohr, B.R. Mottelson, D. Pines, Phys. Rev. 100 936 (1958)

    work page 1958

  5. [5]

    R. A. Broglia and V. V. Zelevinsky, Fifty Years of Nuclear BCS: Pairing in Finite Systems (World Scientific, 2013)

    work page 2013

  6. [6]

    D. J. Dean and M. Hjorth-Jensen, Rev. Mod. Phys. 75 607 (2003)

    work page 2003

  7. [7]

    Frauendorf and A

    S. Frauendorf and A. O. Macchiavelli, Prog. Part. Nucl. P hys. 78 24-90 (2014)

    work page 2014

  8. [8]

    A. M. Lane, Nuclear Theory, Benjamin (1964)

    work page 1964

  9. [9]

    G. F. Bertsch and Y. Luo, Phys. Rev. C 81 064320 (2010)

    work page 2010

  10. [10]

    Sagawa, C

    H. Sagawa, C. L. Bai, and G. Colo, Phys. Scr. 91 083011 (2016)

    work page 2016

  11. [11]

    Kaneko, Y

    K. Kaneko, Y. Sun, T. Mizusaki, Phys. Rev. C 97, 054326 (2018). 12

    work page 2018

  12. [12]

    Zuo, Phys

    U.Lombardo, H.-J.Schulze, and W. Zuo, Phys. Rev. C. 59, 2927 (1999)

    work page 1999

  13. [13]

    Xinle Shang, and Wei Zuo, Phys. Rev. C. 88, 025806 (2013)

    work page 2013

  14. [14]

    T. Alm, G. R¨ opke, and M. Schmidt, Z. Phys. A 337 355 (1990)

    work page 1990

  15. [15]

    Xin-le Shang, Pei Wang, Peng Yin and Wei Zuo, J. Phys. G: N ucl. Part. Phys. 42 055105 (2015)

    work page 2015

  16. [16]

    G.R¨ opke, A.Schnell, P.Schuck, and U.Lombardo, Phys. Rev. C 61, 024306 (2000)

    work page 2000

  17. [17]

    Baldo, U

    M. Baldo, U. Lombardo, H. -J. Schulze, and Zuo Wei, Phys. Rev. C. 66, 054304 (2002)

    work page 2002

  18. [18]

    Bo˙ zek, Phys

    P. Bo˙ zek, Phys. Lett. B 551 93 (2003)

    work page 2003

  19. [19]

    Ø.Elgaroy, L.Engvik, M.Hjorth-Jensen, and E.Osnes, P hys. Rev. C. 57, R1069 (1998)

    work page 1998

  20. [20]

    Clark, C.-G

    J. Clark, C.-G. K¨ allman, C.-H. Yang, D. Chakkalakal, P hys. Lett. B 61 331 (1976)

    work page 1976

  21. [21]

    Wambach, T.L

    J. Wambach, T.L. Ainsworth, D. Pines, Nucl. Phys. A 555 128 (1993)

    work page 1993

  22. [22]

    Schulze, J

    H.-J. Schulze, J. Cugnon, A. Lejeune, M. Baldo, and U. Lo mbardo, Phys. Lett. B 375 1 (1996)

    work page 1996

  23. [23]

    Schulze, A

    H.-J. Schulze, A. Polls, and A. Ramos, Phys. Rev. C 63 044310 (2001)

    work page 2001

  24. [24]

    L. G. Cao, U.Lombardo, P.Schuck, Phys. Rev. C. 74, 064301 (2006)

    work page 2006

  25. [25]

    S. S. Zhang, L. G. Cao, U. Lombardo and P. Schuck, Phys. Re v. C 93 044329 (2016)

    work page 2016

  26. [26]

    lombardo and P

    Wenmei Guo, U. lombardo and P. Schuck, Phys. Rev. C 99 014310 (2019)

    work page 2019

  27. [27]

    Baldo, A

    M. Baldo, A. Grasso, Phys. Lett. B 485 115 (2000)

    work page 2000

  28. [28]

    Lombardo, P

    U. Lombardo, P. Schuck, W. Zuo, Phys. Rev. C 64 021301 (2001)

    work page 2001

  29. [29]

    J. P. Jeukenne, A. Lejeune, and C. Mahaux, Phys. Rep., Ph ys. Lett. bf 25C 83 (1976)

    work page 1976

  30. [30]

    W. Zuo, I. Bombaci, and U. Lombardo, Phys. Rev. C 60 024605 (1999)

    work page 1999

  31. [31]

    Lombardo, P

    U. Lombardo, P. Schuck, and W. Zuo, Phys. Rev. C 64 021301 (R) (2001)

    work page 2001

  32. [32]

    J. M. Dong, U. Lombardo and W. Zuo, Phys. Rev. C 87 062801 (R) (2013)

    work page 2013

  33. [33]

    Grang´ e, A

    P. Grang´ e, A. Lejeune, M. Martzolff, and J. -F. Mathiot, P hys. Rev. C 40 1040 (1989)

    work page 1989

  34. [34]

    W. Zuo, A. Lejeune, U. Lombardo, and J.-F. Mathiot, Nucl . Phys. A 706 418 (2002); Eur. Phys. J. A 14 469 (2002)

    work page 2002

  35. [35]

    H¨ ufner and C

    J. H¨ ufner and C. Mahaux, Ann. Phys. (N.Y.) 73 525 (1972)

    work page 1972

  36. [36]

    Baldo, I

    M. Baldo, I. Bombaci, G. Giansiracusa, U. Lombardo, C. M ahaux and R. Sartor, Phys. Rev. C 41 1748 (1990)

    work page 1990

  37. [37]

    A. A. Abrikosov, L. P. Gorkov, and I. E. Dzyaloshinskii, Methods of Quantum Field Theory in Statistical Physics (Prentice- Hall, Englewood Cliffs, NJ , 1963). 13

    work page 1963

  38. [38]

    Nozi` eres, Theory of Interacting Fermi Systems (Ben jamin, New York, 1966)

    P. Nozi` eres, Theory of Interacting Fermi Systems (Ben jamin, New York, 1966)

    work page 1966

  39. [39]

    A. B. Migdal, Theory of Finite Systems and Applications to Atomic Nuclei (Benjamin, New York, 1964)

    work page 1964

  40. [40]

    J. R. Schrieffer, Theory of Superconductivity (Addison- Wesley, New York, 1964)

    work page 1964

  41. [41]

    Sedrakian and U

    A. Sedrakian and U. Lombardo, Phys. Rev. Lett. 84, 602 (2000)

    work page 2000

  42. [42]

    Sedrakian, Phys

    A. Sedrakian, Phys. Rev. C. 63, 025801 (2001)

    work page 2001

  43. [43]

    Baldo, U

    M. Baldo, U. Lombardo, and P. Schuck, Phys. Rev. C 52 975 (1995)

    work page 1995

  44. [44]

    Bo˙ zek, Phys

    P. Bo˙ zek, Phys. Rev. C 62 054316 (2000)

    work page 2000

  45. [45]

    Lombardo, P

    Caiwan Shen, U. Lombardo, P. Schuck, W. Zuo, and N. Sandu lescu, Phys. Rev. C 67 061302 (R) (2003)

    work page 2003

  46. [46]

    G. F. Bertsch and H. Esbensen, Ann. Phys. (N.Y.) 209, 327 (1991); H. Esbensen, G. F. Bertsch, and K. Hencken, Phys. Rev. C 56, 3054 (1997)

    work page 1991

  47. [47]

    Lombardo, P

    Caiwan Shen, U. Lombardo, P. Schuck, Phys. Rev. C 71 054301 (R) (2005). 14

    work page 2005