- Physical Review Journals
All JournalsPhysics Magazine

Physical Review C

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Bayesian approach for many-body uncertainties in nuclear structure: Many-body perturbation theory for finite nuclei

I. Svensson*, A. Tichai†, K. Hebeler‡, and A. Schwenk§

  • *Contact author: isak.svensson@tu-darmstadt.de
  • †Contact author: alexander.tichai@tu-darmstadt.de
  • ‡Contact author: kai.hebeler@physik.tu-darmstadt.de
  • §Contact author: schwenk@physik.tu-darmstadt.de
PDF

Phys. Rev. C 113, 024303 – Published 4 February, 2026

DOI: https://doi.org/10.1103/y8kt-mgf5

Abstract

A comprehensive assessment of theoretical uncertainties defines an important frontier in nuclear structure research. Ideally, theory predictions include uncertainty estimates that take into account truncation effects from both the interactions and the many-body expansion. While the uncertainties from the expansion of the interactions within effective field theories have been studied systematically using Bayesian methods, many-body truncations are usually addressed by expert assessment. In this work we use a Bayesian framework to study many-body uncertainties within many-body perturbation theory applied to finite nuclei. Our framework is applied to a broad range of nuclei across the nuclear chart calculated from two- and three-nucleon interactions based on chiral effective field theory. These developments represent a step towards a more complete and systematic quantification of uncertainties in ab initio calculations of nuclei.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (63)

  1. H. Hergert, A guided tour of ab initio nuclear many-body theory, Front. Phys. 8, 379 (2020).
  2. K. Hebeler, Three-Nucleon forces: Implementation and applications to atomic nuclei and dense matter, Phys. Rep. 890, 1 (2021).
  3. S. R. Stroberg, J. D. Holt, A. Schwenk, and J. Simonis, Ab initio limits of atomic nuclei, Phys. Rev. Lett. 126, 022501 (2021).
  4. A. Tichai, P. Demol, and T. Duguet, Towards heavy-mass ab initio nuclear structure: Open-shell Ca, Ni and Sn isotopes from Bogoliubov coupled-cluster theory, Phys. Lett. B 851, 138571 (2024).
  5. B. Hu, W. Jiang, T. Miyagi, Z. Sun, A. Ekström, C. Forssén, G. Hagen, J. D. Holt, T. Papenbrock, S. R. Stroberg, and I. Vernon, Ab initio predictions link the neutron skin of Pb208 to nuclear forces, Nat. Phys. 18, 1196 (2021).
  6. T. Miyagi, X. Cao, R. Seutin, S. Bacca, R. F. Garcia Ruiz, K. Hebeler, J. D. Holt, and A. Schwenk, Impact of two-body currents on magnetic dipole moments of nuclei, Phys. Rev. Lett. 132, 232503 (2024).
  7. P. Arthuis, K. Hebeler, and A. Schwenk, Neutron-rich nuclei and neutron skins from chiral low-resolution interactions, arXiv:2401.06675.
  8. G. Hagen, S. J. Novario, Z. H. Sun, T. Papenbrock, G. R. Jansen, J. G. Lietz, T. Duguet, and A. Tichai, Angular-momentum projection in coupled-cluster theory: Structure of Mg34, Phys. Rev. C 105, 064311 (2022).
  9. M. Frosini, T. Duguet, J.-P. Ebran, B. Bally, T. Mongelli, T. R. Rodríguez, R. Roth, and V. Somà, Multi-reference many-body perturbation theory for nuclei: II. Ab initio study of neon isotopes via PGCM and IM-NCSM calculations, Eur. Phys. J. A 58, 63 (2022).
  10. Z. H. Sun, A. Ekström, C. Forssén, G. Hagen, G. R. Jansen, and T. Papenbrock, Multiscale physics of atomic nuclei from first principles, Phys. Rev. X 15, 011028 (2025).
  11. V. Cirigliano, Z. Davoudi, J. Engel, R. J. Furnstahl, G. Hagen, U. Heinz, H. Hergert, M. Horoi, C. Johnson, A. Lovato, E. Mereghetti, W. Nazarewicz, A. Nicholson, T. Papenbrock, S. Pastore, M. Plumlee, D. R. Phillips, P. E. Shanahan, S. R. Stroberg, F. Viens, et al., Towards precise and accurate calculations of neutrinoless double-beta decay, J. Phys. G: Nucl. Part. Phys. 49, 120502 (2022).
  12. A. Belley, J. M. Yao, B. Bally, J. Pitcher, J. Engel, H. Hergert, J. D. Holt, T. Miyagi, T. R. Rodriguez, A. M. Romero, S. R. Stroberg, and X. Zhang, Ab initio uncertainty quantification of neutrinoless double-beta decay in Ge76, Phys. Rev. Lett. 132, 182502 (2024).
  13. M. Door, C.-H. Yeh, M. Heinz, F. Kirk, C. Lyu, T. Miyagi, J. C. Berengut, J. Bieroń, K. Blaum, L. S. Dreissen, S. Eliseev, P. Filianin, M. Filzinger, E. Fuchs, H. A. Fürst, G. Gaigalas, Z. Harman, J. Herkenhoff, N. Huntemann, C. H. Keitel, et al., Search for new bosons with ytterbium isotope shifts, Phys. Rev. Lett. 134, 063002 (2025).
  14. C. Drischler, K. Hebeler, and A. Schwenk, Chiral interactions up to next-to-next-to-next-to-leading order and nuclear saturation, Phys. Rev. Lett. 122, 042501 (2019).
  15. F. Marino, W. Jiang, and S. J. Novario, Diagrammatic ab initio methods for infinite nuclear matter with modern chiral interactions, Phys. Rev. C 110, 054322 (2024).
  16. I. Tews, R. Somasundaram, D. Lonardoni, H. Göttling, R. Seutin, J. Carlson, S. Gandolfi, K. Hebeler, and A. Schwenk, Neutron matter from local chiral effective field theory interactions at large cutoffs, Phys. Rev. Res. 7, 033024 (2025).
  17. C. L. Armstrong, P. Giuliani, K. Godbey, R. Somasundaram, and I. Tews, Emulators for scarce and noisy data: Application to auxiliary-field diffusion Monte Carlo for neutron matter, Phys. Rev. Lett. 135, 142501 (2025).
  18. J. Keller, K. Hebeler, and A. Schwenk, Nuclear equation of state for arbitrary proton fraction and temperature based on chiral effective field theory and a Gaussian process emulator, Phys. Rev. Lett. 130, 072701 (2023).
  19. E. Epelbaum, H.-W. Hammer, and U.-G. Meißner, Modern theory of nuclear forces, Rev. Mod. Phys. 81, 1773 (2009).
  20. R. Machleidt and D. R. Entem, Chiral effective field theory and nuclear forces, Phys. Rep. 503, 1 (2011).
  21. G. Hagen, T. Papenbrock, M. Hjorth-Jensen, and D. J. Dean, Coupled-cluster computations of atomic nuclei, Rep. Prog. Phys. 77, 096302 (2014).
  22. H. Hergert, S. K. Bogner, T. D. Morris, A. Schwenk, and K. Tsukiyama, The in-medium similarity renormalization group: A novel ab initio method for nuclei, Phys. Rep. 621, 165 (2016).
  23. S. R. Stroberg, H. Hergert, S. K. Bogner, and J. D. Holt, Nonempirical interactions for the nuclear shell model: An update, Annu. Rev. Nucl. Part. Sci. 69, 307 (2019).
  24. C. Barbieri and A. Carbone, Self-consistent Green's function approaches, in An Advanced Course in Computational Nuclear Physics: Bridging the Scales from Quarks to Neutron Stars, edited by M. Hjorth-Jensen, M. P. Lombardo, and U. van Kolck (Springer International Publishing, Cham, 2017), p. 571.
  25. V. Somà, P. Navrátil, F. Raimondi, C. Barbieri, and T. Duguet, Novel chiral Hamiltonian and observables in light and medium-mass nuclei, Phys. Rev. C 101, 014318 (2020).
  26. A. Tichai, R. Roth, and T. Duguet, Many-Body perturbation theories for finite nuclei, Front. Phys. 8, 164 (2020).
  27. J. Carlson, S. Gandolfi, F. Pederiva, S. C. Pieper, R. Schiavilla, K. E. Schmidt, and R. B. Wiringa, Quantum Monte Carlo methods for nuclear physics, Rev. Mod. Phys. 87, 1067 (2015).
  28. B. R. Barrett, P. Navrátil, and J. P. Vary, Ab initio no core shell model, Prog. Part. Nucl. Phys. 69, 131 (2013).
  29. R. Roth, Importance truncation for large-scale configuration interaction approaches, Phys. Rev. C 79, 064324 (2009).
  30. R. J. Furnstahl, N. Klco, D. R. Phillips, and S. Wesolowski, Quantifying truncation errors in effective field theory, Phys. Rev. C 92, 024005 (2015).
  31. J. A. Melendez, S. Wesolowski, and R. J. Furnstahl, Bayesian truncation errors in chiral effective field theory: Nucleon-nucleon observables, Phys. Rev. C 96, 024003 (2017).
  32. S. Wesolowski, R. J. Furnstahl, J. A. Melendez, and D. R. Phillips, Exploring Bayesian parameter estimation for chiral effective field theory using nucleon–nucleon phase shifts, J. Phys. G 46, 045102 (2019).
  33. J. Hoppe, C. Drischler, K. Hebeler, A. Schwenk, and J. Simonis, Probing chiral interactions up to next-to-next-to-next-to-leading order in medium-mass nuclei, Phys. Rev. C 100, 024318 (2019).
  34. J. A. Melendez, R. J. Furnstahl, D. R. Phillips, M. T. Pratola, and S. Wesolowski, Quantifying correlated truncation errors in effective field theory, Phys. Rev. C 100, 044001 (2019).
  35. T. Hüther, K. Vobig, K. Hebeler, R. Machleidt, and R. Roth, Family of chiral two- plus three-nucleon interactions for accurate nuclear structure studies, Phys. Lett. B 808, 135651 (2020).
  36. C. Drischler, R. J. Furnstahl, J. A. Melendez, and D. R. Phillips, How well do we know the neutron-matter equation of state at the densities inside neutron stars? A Bayesian approach with correlated uncertainties, Phys. Rev. Lett. 125, 202702 (2020).
  37. P. Maris, E. Epelbaum, R. Furnstahl, J. Golak, K. Hebeler, T. Hüther, H. Kamada, H. Krebs, U.-G. Meißner, J. A. Melendez, A. Nogga, P. Reinert, R. Roth, R. Skibiński, V. Soloviov, K. Topolnicki, J. P. Vary, Y. Volkotrub, H. Witała, and T. Wolfgruber, Light nuclei with semilocal momentum-space regularized chiral interactions up to third order, Phys. Rev. C 103, 054001 (2021).
  38. S. Wesolowski, I. Svensson, A. Ekström, C. Forssén, R. J. Furnstahl, J. A. Melendez, and D. R. Phillips, Rigorous constraints on three-nucleon forces in chiral effective field theory from fast and accurate calculations of few-body observables, Phys. Rev. C 104, 064001 (2021).
  39. I. Svensson, A. Ekström, and C. Forssén, Bayesian parameter estimation in chiral effective field theory using the Hamiltonian Monte Carlo method, Phys. Rev. C 105, 014004 (2022).
  40. I. Svensson, A. Ekström, and C. Forssén, Bayesian estimation of the low-energy constants up to fourth order in the nucleon-nucleon sector of chiral effective field theory, Phys. Rev. C 107, 014001 (2023).
  41. W. G. Jiang, C. Forssén, T. Djärv, and G. Hagen, Nuclear-matter saturation and symmetry energy within Δ-full chiral effective field theory, Phys. Rev. C 109, L061302 (2024).
  42. I. Svensson, A. Ekström, and C. Forssén, Inference of the low-energy constants in Δ-full chiral effective field theory including a correlated truncation error, Phys. Rev. C 109, 064003 (2024).
  43. P. J. Millican, R. J. Furnstahl, J. A. Melendez, D. R. Phillips, and M. T. Pratola, Assessing correlated truncation errors in modern nucleon-nucleon potentials, Phys. Rev. C 110, 044002 (2024).
  44. I. Svensson, A. Tichai, K. Hebeler, and A. Schwenk, Reproduction package: A Bayesian approach for many-body uncertainties in nuclear structure: Many-body perturbation theory for finite nuclei (v1.0), Zenodo (2025), doi:10.5281/zenodo.17192609.
  45. https://github.com/svisak/manybody_uncertainties.
  46. I. Shavitt and R. J. Bartlett, Many-Body Methods in Chemistry and Physics: MBPT and Coupled-Cluster Theory, Cambridge Molecular Science (Cambridge University Press, Cambridge, 2009).
  47. J. Goldstone, Derivation of the brueckner many-body theory, Proc. Roy. Soc. A 239, 267 (1957).
  48. T. Miyagi, S. R. Stroberg, P. Navrátil, K. Hebeler, and J. D. Holt, Converged ab initio calculations of heavy nuclei, Phys. Rev. C 105, 014302 (2022).
  49. A. Tichai, J. Langhammer, S. Binder, and R. Roth, Hartree-Fock many-body perturbation theory for nuclear ground-states, Phys. Lett. B 756, 283 (2016).
  50. F. Alp, Y. Dietz, K. Hebeler, and A. Schwenk, Equation of state and Fermi liquid properties of dense matter based on chiral effective field theory interactions, Phys. Rev. C 112, 055802 (2025).
  51. K. Hebeler, S. K. Bogner, R. J. Furnstahl, A. Nogga, and A. Schwenk, Improved nuclear matter calculations from chiral low-momentum interactions, Phys. Rev. C 83, 031301(R) (2011).
  52. T. Miyagi (private communication).
  53. S. Wesolowski, N. Klco, R. J. Furnstahl, D. R. Phillips, and A. Thapaliya, Bayesian parameter estimation for effective field theories, J. Phys. G 43, 074001 (2016).
  54. C. R. Harris, K. J. Millman, S. J. van der Walt, R. Gommers, P. Virtanen, D. Cournapeau, E. Wieser, J. Taylor, S. Berg, N. J. Smith, R. Kern, M. Picus, S. Hoyer, M. H. van Kerkwijk, M. Brett, A. Haldane, J. F. del Río, M. Wiebe, P. Peterson, P. Gérard-Marchant, et al., Array programming with NumPy, Nature (London) 585, 357 (2020).
  55. D. Foreman-Mackey, D. W. Hogg, D. Lang, and J. Goodman, Emcee: The MCMC hammer, Publ. Astron. Soc. Pac. 125, 306 (2013).
  56. W. G. Jiang, A. Ekström, C. Forssén, G. Hagen, G. R. Jansen, and T. Papenbrock, Accurate bulk properties of nuclei from A=2 to ∞ from potentials with Δ isobars, Phys. Rev. C 102, 054301 (2020).
  57. A. Ekström, G. R. Jansen, K. A. Wendt, G. Hagen, T. Papenbrock, B. D. Carlsson, C. Forssén, M. Hjorth-Jensen, P. Navrátil, and W. Nazarewicz, Accurate nuclear radii and binding energies from a chiral interaction, Phys. Rev. C 91, 051301(R) (2015).
  58. M. Heinz, T. Miyagi, S. R. Stroberg, A. Tichai, K. Hebeler, and A. Schwenk, Improved structure of calcium isotopes from ab initio calculations, Phys. Rev. C 111, 034311 (2025).
  59. A. Tichai, E. Gebrerufael, K. Vobig, and R. Roth, Open-shell nuclei from no-core shell model with perturbative improvement, Phys. Lett. B 786, 448 (2018).
  60. A. Tichai, P. Arthuis, T. Duguet, H. Hergert, V. Somá, and R. Roth, Bogoliubov many-body perturbation theory for open-shell nuclei, Phys. Lett. B 786, 195 (2018).
  61. M. Frosini, T. Duguet, J.-P. Ebran, B. Bally, H. Hergert, T. R. Rodríguez, R. Roth, J. Yao, and V. Somà, Multi-reference many-body perturbation theory for nuclei: III. Ab initio calculations at second order in PGCM-PT, Eur. Phys. J. A 58, 64 (2022).
  62. P. Demol, M. Frosini, A. Tichai, V. Somà, and T. Duguet, Bogoliubov many-body perturbation theory under constraint, Ann. Phys. 424, 168358 (2021).
  63. P. Demol, T. Duguet, and A. Tichai, Ab initio Bogoliubov many-body perturbation theory: Closed-form constraint on the average particle number, Eur. Phys. J. A 61, 16 (2025).

Outline

Information

Phys. Rev. C 113, 024303– Published 4 February, 2026
  • Received 22 July 2025
  • Revised 26 September 2025
  • Accepted 17 November 2025
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation