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arxiv: 2511.08366 · v2 · pith:YKVIHAJXnew · submitted 2025-11-11 · ⚛️ nucl-th

Finite-range pairing in nuclear density functional theory

Sudhanva Lalit , Paul-Gerhard Reinhard , Kyle Godbey , Witold Nazarewicz This is my paper

Pith reviewed 2026-05-22 12:05 UTC · model grok-4.3

classification ⚛️ nucl-th
keywords nuclear density functional theorypairing correlationsfinite-range pairingGaussian foldingcontinuum statesnuclear structurepair density
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The pith

Folding pair densities with Gaussians of radius near 1 fm stabilizes pairing calculations in nuclear DFT.

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

Zero-range pairing functionals in nuclear density functional theory produce divergences and instabilities when single-particle spaces grow large enough to include unbound continuum states. The paper remedies this by replacing zero-range interactions with finite-range versions constructed through Gaussian folding of the pair density. Calculations indicate that a folding radius of about 1 fm gives the best balance between preserving accuracy in bound-state observables and eliminating numerical pathologies across different applications. This regularization works even for functionals that include gradient-density dependence, such as the Fayans form.

Core claim

Folding the pair density with a Gaussian of finite range yields a pairing functional whose results become independent of the size of the single-particle basis once the folding radius is chosen near 1 fm, thereby removing the continuum divergences that plague zero-range functionals while leaving the quality of bound-state predictions essentially unchanged.

What carries the argument

Gaussian folding of the pair density, which converts a zero-range pairing interaction into a finite-range one that regularizes the pairing field against continuum divergences.

If this is right

  • Pairing calculations become stable and basis-size independent even in very large spaces containing many continuum states.
  • Gradient-dependent pairing functionals such as Fayans can be used reliably without strong cutoff dependence.
  • Global surveys of nuclear properties gain robustness because results no longer hinge on arbitrary choices for marginally occupied states.
  • The same regularization can be applied to other observables that involve pair densities without introducing new parameters.

Where Pith is reading between the lines

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

  • The 1 fm scale may connect to the effective range of the underlying nucleon-nucleon force and could be tested by varying the folding function shape.
  • The approach might extend naturally to time-dependent density functional calculations or to pairing in neutron-star crust matter where continuum effects are large.
  • Comparison with ab initio pairing results in medium-mass nuclei could check whether the regularization preserves microscopic accuracy.

Load-bearing premise

The Gaussian folding at a radius near 1 fm keeps the essential short-range pairing physics intact while removing the divergences.

What would settle it

Numerical results with the folded functional that deviate markedly from converged zero-range results in small model spaces or from experimental pairing gaps would indicate the regularization has altered the physics.

Figures

Figures reproduced from arXiv: 2511.08366 by Kyle Godbey, Paul-Gerhard Reinhard, Sudhanva Lalit, Witold Nazarewicz.

Figure 1
Figure 1. Figure 1: FIG. 1. Cut of the density distributions along [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Density of neutron s.p. states in [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. The global quality measure [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Trends of key observables for BCS with [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. (a) Charge radius and (b) total energy versus size of [PITH_FULL_IMAGE:figures/full_fig_p007_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. (a) Charge radius and (b) total energy as a function [PITH_FULL_IMAGE:figures/full_fig_p008_8.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Density of neutron s.p. states in [PITH_FULL_IMAGE:figures/full_fig_p009_10.png] view at source ↗
read the original abstract

Pairing correlations are ubiquitous in low-energy states of atomic nuclei. To incorporate them within nuclear density functional theory, often used for global computations of nuclear properties, pairing functionals that generate nucleonic pair densities and pairing fields are introduced. Many pairing functionals currently used can be traced back to zero-range nucleon-nucleon interactions. Unfortunately, such functionals are plagued by deficiencies that become apparent in large model spaces that contain unbound single-particle (continuum) states. In particular, the underlying computational schemes diverge as the single-particle space increases, and the results depend on how marginally occupied states are incorporated. These problems become more pronounced for pairing functionals that contain gradient-density dependence, such as in the Fayans functional. To remedy this, finite-range pairing functionals are introduced. In this study, this is done by folding the pair density with Gaussians. We show that a folding radius of about 1\,fm offers the best compromise between quality and stability, and substantially reduces the pathological behavior in different numerical applications.

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 manuscript proposes finite-range pairing functionals for nuclear density functional theory by folding the pair density with a Gaussian to address divergences and numerical instabilities that arise with zero-range interactions in large model spaces containing continuum states. It focuses on mitigating problems that are exacerbated in gradient-dependent functionals such as the Fayans form and concludes that a folding radius of approximately 1 fm provides the optimal compromise between quality and numerical stability while substantially reducing pathological behaviors across applications.

Significance. If the numerical demonstrations confirm that bound-state observables remain essentially unchanged while stability improves, the approach would offer a low-overhead practical fix for continuum-related issues in global DFT calculations, adding only a single parameter without requiring a full microscopic finite-range force. This could enhance reliability of pairing predictions in extended bases and reduce sensitivity to cutoff choices.

major comments (2)
  1. [Abstract] Abstract: the assertion that a folding radius of about 1 fm 'offers the best compromise between quality and stability' and 'substantially reduces the pathological behavior' is presented without any quantitative tables, error estimates, or explicit comparisons to zero-range results, leaving the central claim without visible supporting data in the summary of findings.
  2. [Folding procedure and numerical tests] The folding construction (direct Gaussian smearing of the pair density rather than derivation from a finite-range force): because only the radius is added and no compensating readjustment of the pairing strength is introduced, any alteration of the effective pairing field or its coupling to density gradients must be shown not to shift gaps or odd-even staggering even in small, fully convergent bases; explicit verification of equivalence for bound-state observables is required to support the claim that essential physics is preserved.
minor comments (1)
  1. Clarify in the text how the Gaussian folding is implemented numerically (e.g., whether it is applied before or after discretization) and ensure all convergence plots explicitly label the model-space sizes and radii tested.

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 below and indicate the revisions we will make.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the assertion that a folding radius of about 1 fm 'offers the best compromise between quality and stability' and 'substantially reduces the pathological behavior' is presented without any quantitative tables, error estimates, or explicit comparisons to zero-range results, leaving the central claim without visible supporting data in the summary of findings.

    Authors: The abstract is a concise summary of conclusions supported by the detailed numerical tests and comparisons in the body of the manuscript, including systematic variations of folding radii, model-space sizes, and direct contrasts with zero-range results shown in the figures and tables of the results section. We agree that the abstract would benefit from a brief quantitative anchor. In the revised manuscript we will update the abstract to reference the observed reduction in pathological cutoff dependence while preserving the overall length and readability. revision: yes

  2. Referee: [Folding procedure and numerical tests] The folding construction (direct Gaussian smearing of the pair density rather than derivation from a finite-range force): because only the radius is added and no compensating readjustment of the pairing strength is introduced, any alteration of the effective pairing field or its coupling to density gradients must be shown not to shift gaps or odd-even staggering even in small, fully convergent bases; explicit verification of equivalence for bound-state observables is required to support the claim that essential physics is preserved.

    Authors: We acknowledge that the Gaussian folding is a direct regularization of the pair density rather than a microscopic finite-range force. In the manuscript we already demonstrate that the integrated pairing strength is preserved and that, for a folding radius of 1 fm, the changes to pairing gaps and odd-even staggering remain within numerical precision in fully convergent small bases. To make this verification more explicit we will add a short dedicated paragraph (or small table) in the revised version that directly compares bound-state observables between the original zero-range functional and the folded version in those small spaces. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper introduces finite-range pairing functionals by folding the pair density with a Gaussian of chosen radius ~1 fm as a direct modification to address continuum divergences in zero-range functionals. The radius is selected numerically for best compromise between quality and stability, with results demonstrated in applications. This construction is presented as an independent ansatz and numerical fix rather than a derivation that reduces by definition or self-citation to the same data or inputs it aims to predict. No load-bearing steps, self-definitional equations, or fitted inputs called predictions are evident that would make central claims equivalent to their inputs by construction. The approach remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim depends on the empirical choice of a single length-scale parameter (folding radius) tuned to balance quality and stability; no new particles or forces are postulated.

free parameters (1)
  • Gaussian folding radius
    Selected near 1 fm as the value offering the best compromise between quality and numerical stability in the tested applications.
axioms (1)
  • domain assumption Standard nuclear density functional theory framework remains valid when the pairing interaction is made finite-range via Gaussian folding.
    Invoked throughout the abstract as the setting in which the new functionals are introduced.

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