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arxiv: 2605.26088 · v2 · pith:KUMWSEYBnew · submitted 2026-05-25 · ⚛️ nucl-th · hep-ph· nucl-ex

Multipole tomography of atomic nuclei with symmetry-conserved theories

X. Sun , J. Dobaczewski , W. Nazarewicz , H. Wibowo This is my paper

Pith reviewed 2026-06-29 19:06 UTC · model grok-4.3

classification ⚛️ nucl-th hep-phnucl-ex
keywords nuclear deformationssymmetry restorationquadrupole momentsdensity functional theorytwo-body correlationsangular momentum projectionintrinsic frameconditional probabilities
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The pith

Two-body conditional probabilities define quadrupole deformations for angular-momentum conserving nuclear states.

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

The paper introduces two-body conditional probabilities of nucleon positions to define an intrinsic reference frame and multipole moments for many-body wave functions that conserve total angular momentum J. This makes it possible to characterize quadrupole shapes in states with J equal to zero or one-half, where one-body spectroscopic moments are zero or inaccessible. The method is illustrated in density functional theory calculations for the J=0 ground states of oxygen-16 and neon-20, where rotational symmetry is restored from prolate or oblate intrinsic shapes. The resulting two-body shape measures are shown to differ from the one-body quadrupole moments extracted from the corresponding broken-symmetry configurations.

Core claim

To define the intrinsic reference frame and multipole moments of angular-momentum-J-conserving many-body wave functions, we introduce two-body conditional probabilities of finding two nucleons at different positions in space. In this way, quadrupole deformations of states with J≤1/2, which are not accessible via spectroscopic one-body quadrupole moments, can be characterized. We illustrate the method with nuclear density functional theory calculations for J=0 states of 16O and 20Ne, the latter obtained by restoring rotational symmetry of prolate or oblate intrinsic configurations. We show that the two-body quadrupole shape characterizations differ from one-body moments obtained from broken-s

What carries the argument

Two-body conditional probabilities of finding two nucleons at different positions in space, used to define an intrinsic reference frame and multipole moments.

If this is right

  • Quadrupole deformations become characterizable for all J=0 states within symmetry-conserving calculations.
  • Shape information can be extracted without relying on auxiliary symmetry-broken mean-field solutions.
  • Direct comparisons between theory and experiment become possible for even-even nuclei using only conserved-symmetry wave functions.
  • The method extends the range of nuclei whose intrinsic shapes can be discussed in angular-momentum projected approaches.

Where Pith is reading between the lines

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

  • The two-body approach could be applied to extract higher-order multipoles or to map full nuclear density distributions from symmetry-conserved states.
  • It may provide a route to reconcile spectroscopic data with ab initio calculations that strictly conserve angular momentum.
  • The conditional-probability construction might generalize to other fermionic systems where intrinsic frames are needed but total angular momentum is a good quantum number.

Load-bearing premise

Two-body conditional probabilities of nucleon positions suffice to define a unique intrinsic reference frame and multipole moments for J-conserving many-body wave functions.

What would settle it

A direct computation in which the two-body conditional probabilities fail to produce a stable, unique set of multipole moments that is independent of the choice of coordinate origin or that contradicts measured transition strengths in a J=0 state.

Figures

Figures reproduced from arXiv: 2605.26088 by H. Wibowo, J. Dobaczewski, W. Nazarewicz, X. Sun.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: shows the cumulative quadrupole moments calculated up to order n = 10. We observe that, at every order, their values differ significantly from the intrinsic moments shown in [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
read the original abstract

To define the intrinsic reference frame and multipole moments of angular-momentum-$J$-conserving many-body wave functions, we introduce two-body conditional probabilities of finding two nucleons at different positions in space. In this way, quadrupole deformations of states with $J\leq1/2$, which are not accessible via spectroscopic one-body quadrupole moments, can be characterized. We illustrate the method with nuclear density functional theory calculations for $J=0$ states of $^{16}$O and $^{20}$Ne, the latter obtained by restoring rotational symmetry of prolate or oblate intrinsic configurations. We show that the two-body quadrupole shape characterizations differ from one-body moments obtained from broken-symmetry states.

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 introduces two-body conditional probabilities of nucleon positions to define intrinsic reference frames and multipole moments for angular-momentum J-conserving many-body wave functions. This enables characterization of quadrupole deformations for states with J ≤ 1/2 that are inaccessible via one-body spectroscopic quadrupole moments. The method is illustrated with nuclear density functional theory calculations for the J=0 ground states of 16O and 20Ne (the latter obtained via rotational symmetry restoration from prolate or oblate intrinsic configurations), showing that the resulting two-body quadrupole characterizations differ from one-body moments extracted from the corresponding broken-symmetry states.

Significance. If the construction is robust, the approach supplies a concrete route to extract intrinsic shape information directly from symmetry-conserved wave functions, a long-standing need in nuclear many-body theory where symmetry restoration is routinely employed. The explicit numerical comparison for 16O and 20Ne demonstrates that the two-body route can yield shape measures distinct from those of the underlying intrinsic states, potentially clarifying the interpretation of deformation in projected calculations.

major comments (1)
  1. [Abstract] Abstract (first sentence) and method section: the central claim that two-body conditional probabilities P(r1,r2) suffice to define a unique intrinsic reference frame for J-conserving states is not accompanied by an explicit demonstration of uniqueness or invariance under re-conditioning. For J=0 states the total wave function is rotationally invariant; without a proof that the conditional density fixes a unique principal-axis choice (or that any ambiguity does not affect the extracted quadrupole tensor), the reported difference from broken-symmetry one-body moments cannot be unambiguously attributed to intrinsic shape rather than frame convention.
minor comments (1)
  1. The explicit mathematical definition of the two-body conditional probability and the procedure for extracting the quadrupole tensor from it should be stated in an equation early in the method section for clarity.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the constructive comment on the uniqueness of the intrinsic frame. We address the point below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Abstract] Abstract (first sentence) and method section: the central claim that two-body conditional probabilities P(r1,r2) suffice to define a unique intrinsic reference frame for J-conserving states is not accompanied by an explicit demonstration of uniqueness or invariance under re-conditioning. For J=0 states the total wave function is rotationally invariant; without a proof that the conditional density fixes a unique principal-axis choice (or that any ambiguity does not affect the extracted quadrupole tensor), the reported difference from broken-symmetry one-body moments cannot be unambiguously attributed to intrinsic shape rather than frame convention.

    Authors: We agree that an explicit demonstration of uniqueness and invariance under re-conditioning is missing from the current text. In the revised manuscript we will insert a short subsection in the method section that (i) fixes one nucleon at a reference point along the z-axis by construction, (ii) shows that the resulting two-body quadrupole tensor is diagonal in the body-fixed frame defined by its eigenvectors, and (iii) proves that re-conditioning on any other nucleon yields a tensor that differs only by a global rotation, which is removed by the same diagonalization. Because the J=0 wave function is rotationally invariant, this procedure selects a unique principal-axis system (up to the discrete 180° ambiguities inherent to quadrupole shapes). The added argument will make clear that the reported difference with broken-symmetry one-body moments originates in the two-body correlations rather than in an arbitrary frame choice. revision: yes

Circularity Check

0 steps flagged

No significant circularity; new definition applied to independent calculations

full rationale

The paper defines the intrinsic frame via introduced two-body conditional probabilities and then applies this to DFT calculations on specific nuclei (16O, 20Ne) to exhibit differences from broken-symmetry one-body moments. No quoted equations, self-citations, or fitted parameters reduce the reported characterizations to tautological inputs by construction. The central illustration rests on explicit numerical results rather than renaming or self-referential fitting, rendering the derivation self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; ledger left empty.

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Forward citations

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