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arxiv: 2607.00119 · v1 · pith:KOEGTNZ2 · submitted 2026-06-30 · nucl-th

High-order perturbative calculations of nuclear ground states: Automated evaluation of many-body diagrams

Zhen Li , Alexander Tichai , Achim Schwenk , Nadezda A. Smirnova This is my paper

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classification nucl-th
keywords many-body perturbation theorynuclear ground statesdiagram automationchiral effective field theoryclosed-shell nucleiconvergence trendsIMSRG comparisonfifth-order calculations
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The pith

Automated diagram evaluation extends many-body perturbation theory for nuclear ground states to fifth order.

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

The paper shows that many-body perturbation theory calculations of ground-state energies and radii in closed-shell nuclei can be pushed beyond third order by automating the generation and evaluation of all diagrams through fifth order. Using two- and three-nucleon forces from chiral effective field theory, the method is applied to nuclei up to nickel-78 and produces a clear convergence pattern in the energies. Fourth-order contributions are typically less than half the size of third-order ones, accompanied by cancellations among diagram classes. The work also compares these perturbative results directly to in-medium similarity renormalization group calculations to illuminate truncation effects in the latter approach.

Core claim

Using automated diagram generation and evaluation up to fifth order, the many-body perturbation theory calculations of the ground-state energy and radius of closed-shell nuclei are advanced beyond third order. For nuclei up to 78Ni with chiral effective field theory interactions, a clear convergence trend is observed for the ground-state energy. The magnitude of the fourth-order contribution is typically less than half of the third order, with typical cancellations among different classes of diagrams. A comprehensive comparison with non-perturbative in-medium similarity renormalization group calculations provides insight into many-body uncertainties associated with the IMSRG(2) truncation.

What carries the argument

Automated generation and evaluation of many-body diagrams in perturbation theory up to fifth order.

If this is right

  • Ground-state energies display a convergence trend that supports calculations at higher accuracy.
  • The fourth-order contribution to the energy is typically less than half the third-order term.
  • Cancellations among different classes of diagrams occur at fourth order.
  • Comparison with IMSRG calculations illuminates many-body uncertainties from the IMSRG(2) truncation.

Where Pith is reading between the lines

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

  • The automation could be extended to compute additional observables such as charge radii or transition strengths at the same orders.
  • The observed pattern of cancellations might allow selective inclusion of diagram classes in future approximations without full fifth-order evaluation.
  • Applying the same automated framework to other effective interactions would test whether the convergence trend persists beyond the chiral forces used here.

Load-bearing premise

The automated diagram generation and evaluation procedure correctly enumerates and computes all many-body diagrams through fifth order without omissions or numerical errors for the chosen chiral interactions and model spaces.

What would settle it

Direct comparison of the automated fourth-order energy for oxygen-16 against an independent manual enumeration of every diagram class with the same interaction and model space would confirm or refute the automation's completeness.

Figures

Figures reproduced from arXiv: 2607.00119 by Achim Schwenk, Alexander Tichai, Nadezda A. Smirnova, Zhen Li.

Figure 1
Figure 1. Figure 1: FIG. 1. Third order Feynman diagrams in MBPT. Upward (downward) solid lines refer to particle (hole) lines. The one-body [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Required computing time as a function of mass num [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Convergence behavior with the multilevel truncation scheme (left) and the homogeneous truncation scheme (middle), [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. MBPT results for the ground-state energies of selected closed-shell nuclei with the 1.8/2.0 (EM) interaction [ [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. MBPT results for charge radii (top), neutron radii (middle), and neutron skins (bottom panels) for [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. MBPT results for the ground-state energy (top), charge radius (middle), and neutron radius of [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Contributions from singles, doubles, triples, and [PITH_FULL_IMAGE:figures/full_fig_p010_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Contributions from singles, doubles, triples, and [PITH_FULL_IMAGE:figures/full_fig_p010_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Contributions from singles, doubles, triples, and [PITH_FULL_IMAGE:figures/full_fig_p010_11.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. Sum of [PITH_FULL_IMAGE:figures/full_fig_p012_13.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. MBPT results with different groups of diagrams [PITH_FULL_IMAGE:figures/full_fig_p012_12.png] view at source ↗
read the original abstract

We advance the many-body perturbation theory (MBPT) calculations of the ground-state energy and radius of closed-shell nuclei beyond third order. Using automated diagram generation and evaluation up to fifth order, we present ground-state properties of selected closed-shell nuclei up to $^{78}$Ni with two- and three-nucleon interactions derived from chiral effective field theory. A clear convergence trend is observed for the ground-state energy enabling calculations at improved accuracy. We further investigate in detail the decomposition of the fourth-order contributions. For the ground-state energy, the magnitude of the fourth-order contribution is typically less than half of the third order, and a typical cancellation among different classes of diagrams is observed. Finally, we perform a comprehensive comparison between MBPT and non-perturbative in-medium similarity renormalization group (IMSRG) calculations, with the goal to provide insight into many-body uncertainties associated with the IMSRG(2) truncation.

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 / 0 minor

Summary. The paper advances many-body perturbation theory (MBPT) calculations of ground-state energies and radii for closed-shell nuclei beyond third order by using automated diagram generation and evaluation up to fifth order with chiral EFT two- and three-nucleon interactions. It reports a clear convergence trend for the ground-state energy, with the fourth-order contribution typically less than half the third-order magnitude and cancellations among diagram classes, and performs comparisons to IMSRG(2) to assess many-body uncertainties.

Significance. If the automated procedure is shown to be complete and accurate, the work would enable systematic higher-order perturbative calculations in nuclear many-body theory, providing quantitative benchmarks for convergence and many-body truncation errors that complement non-perturbative approaches like IMSRG. The detailed fourth-order decomposition adds insight into diagram cancellations.

major comments (1)
  1. [Abstract] Abstract: The reported convergence trend (fourth-order magnitude typically < half of third-order, with cancellations) and subsequent IMSRG comparison rest on the automated enumeration and summation of all diagrams through fifth order. No benchmarks are described, such as recovery of established third-order MBPT results for the same interactions or agreement with independent manual calculations on a small system (e.g., ^{4}He), leaving open the possibility of systematic omissions that would directly affect the central numerical claims.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comment. We address the major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The reported convergence trend (fourth-order magnitude typically < half of third-order, with cancellations) and subsequent IMSRG comparison rest on the automated enumeration and summation of all diagrams through fifth order. No benchmarks are described, such as recovery of established third-order MBPT results for the same interactions or agreement with independent manual calculations on a small system (e.g., ^{4}He), leaving open the possibility of systematic omissions that would directly affect the central numerical claims.

    Authors: We agree that explicit benchmarks validating the automated diagram enumeration and summation are essential to support the reported results. The current manuscript does not include such benchmarks. In the revised version we will add a dedicated subsection (or appendix) that recovers established third-order MBPT results for the same chiral EFT interactions and compares the automated fifth-order implementation against independent manual calculations for ^{4}He. This addition will directly address the concern about possible systematic omissions. revision: yes

Circularity Check

0 steps flagged

No circularity: results are direct evaluations of diagrams from chiral interactions

full rationale

The paper advances MBPT by automating diagram generation and evaluation up to fifth order for ground-state energies and radii of closed-shell nuclei using chiral EFT interactions. The reported convergence trends (fourth-order magnitude typically < half of third-order, with cancellations) and IMSRG comparisons follow directly from these explicit computations in chosen model spaces. No equations define a quantity in terms of itself, no fitted parameters are relabeled as predictions, and no load-bearing claims rest on self-citations that themselves reduce to the present work. The derivation chain is self-contained against external benchmarks (chiral interactions and model spaces are inputs, not outputs).

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claims rest on the correctness of the automated diagram machinery and the suitability of the input chiral EFT interactions; both are taken from prior literature without new independent checks visible in the abstract.

free parameters (1)
  • chiral EFT low-energy constants
    Parameters of the two- and three-nucleon interactions are determined in earlier chiral EFT fits and enter as fixed inputs.
axioms (1)
  • domain assumption The perturbative expansion of the nuclear Hamiltonian remains valid and systematically improvable for closed-shell nuclei with the chosen interactions and model spaces.
    Invoked when extending the calculation from third to fifth order and claiming convergence.

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discussion (0)

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Reference graph

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