- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review C

Export citation

Export citation

Choose format for download:

Download Citation
  • Editors' Suggestion
  • Access by Xinjiang University

Perturbative effective-field-theory calculation of the deuteron longitudinal response function

Andrew J. Andis1,*, Songlin Lyu (吕松林)2,3, Bingwei Long (龙炳蔚)4,5, and Sebastian König1,†

  • *Contact author: ajandis@ncsu.edu
  • †Contact author: skoenig@ncsu.edu
PDF

Phys. Rev. C 114, 024004 – Published 31 August, 2026

DOI: https://doi.org/10.1103/zgbl-rffb

Abstract

In this work, we study the longitudinal response function of the deuteron up to next-to-next-to-leading order in chiral effective field theory (chiral EFT). We use an approach that maintains exact renormalization group (RG) invariance at each order of the EFT expansion by treating all subleading corrections in perturbation theory. To that end, we extend the Lorentz integral transform method to allow for such a perturbative treatment. In doing so, we further develop the existing work on strictly RG-invariant chiral EFT, which has so far focused primarily on binding energies and static properties, to inelastic processes. We carefully analyze the convergence properties of the theory and find good agreement with available experimental data. Our findings provide the foundation for similar studies of inelastic processes in a range of nuclei, based on perturbatively renormalized EFT schemes.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (75)

  1. H.-W. Hammer, S. König, and U. van Kolck, Nuclear effective field theory: Status and perspectives, Rev. Mod. Phys. 92, 025004 (2020).
  2. S. Weinberg, Nuclear forces from chiral Lagrangians, Phys. Lett. B 251, 288 (1990).
  3. S. Weinberg, Effective chiral Lagrangians for nucleon-pion interactions and nuclear forces, Nucl. Phys. B 363, 3 (1991).
  4. C. Ordóñez and U. van Kolck, Chiral Lagrangians and nuclear forces, Phys. Lett. B 291, 459 (1992).
  5. S. Weinberg, Three-body interactions among nucleons and pions, Phys. Lett. B 295, 114 (1992).
  6. U. L. Van Kolck, Soft Physics: Applications of effective chiral lagrangians to nuclear physics and quark models, Ph.D. thesis, Texas University, 1993.
  7. M. Rho, Exchange currents from chiral Lagrangians, Phys. Rev. Lett. 66, 1275 (1991).
  8. T.-S. Park, D.-P. Min, and M. Rho, Chiral dynamics and heavy-fermion formalism in nuclei: Exchange axial currents, Phys. Rep. 233, 341 (1993).
  9. T.-S. Park, D.-P. Min, and M. Rho, Chiral Lagrangian approach to exchange vector currents in nuclei, Nucl. Phys. A 596, 515 (1996).
  10. S. Kölling, E. Epelbaum, H. Krebs, and U.-G. Meißner, Two-pion exchange electromagnetic current in chiral effective field theory using the method of unitary transformation, Phys. Rev. C 80, 045502 (2009).
  11. S. Kölling, E. Epelbaum, H. Krebs, and U.-G. Meißner, Two-nucleon electromagnetic current in chiral effective field theory: One-pion exchange and short-range contributions, Phys. Rev. C 84, 054008 (2011).
  12. S. Pastore, L. Girlanda, R. Schiavilla, and M. Viviani, Two-nucleon electromagnetic charge operator in chiral effective field theory (χEFT) up to one loop, Phys. Rev. C 84, 024001 (2011).
  13. A. Baroni, L. Girlanda, S. Pastore, R. Schiavilla, and M. Viviani, Nuclear axial currents in chiral effective field theory, Phys. Rev. C 93, 015501 (2016); 93, 049902(E) (2016); 95, 059901(E) (2017).
  14. H. Krebs, Nuclear currents in chiral effective field theory, Eur. Phys. J. A 56, 234 (2020).
  15. M. Hoferichter, P. Klos, and A. Schwenk, Chiral power counting of one- and two-body currents in direct detection of dark matter, Phys. Lett. B 746, 410 (2015).
  16. V. Cirigliano, W. Dekens, J. de Vries, M. L. Graesser, and E. Mereghetti, A neutrinoless double beta decay master formula from effective field theory, J. High Energy Phys. 12 (2018) 097.
  17. F. Oosterhof, B. Long, J. de Vries, R. G. E. Timmermans, and U. van Kolck, Baryon-number violation by two units and the deuteron lifetime, Phys. Rev. Lett. 122, 172501 (2019).
  18. T.-X. Liu, R. Peng, S. Lyu, and B. Long, Renormalization of proton-proton fusion in chiral effective field theory, Phys. Rev. C 106, 055501 (2022).
  19. A. Nogga, R. G. E. Timmermans, and U. van Kolck, Renormalization of one-pion exchange and power counting, Phys. Rev. C 72, 054006 (2005).
  20. M. Pavon Valderrama, Perturbative renormalizability of chiral two-pion exchange in nucleon-nucleon scattering: P and D waves, Phys. Rev. C 84, 064002 (2011).
  21. B. Long and C. J. Yang, Renormalizing chiral nuclear forces: A case study of P03, Phys. Rev. C 84, 057001 (2011).
  22. W. Shi, R. Peng, T.-X. Liu, S. Lyu, and B. Long, Perturbative calculations of deuteron form factors, Phys. Rev. C 106, 015505 (2022).
  23. B. Long and C. J. Yang, Renormalizing chiral nuclear forces: Triplet channels, Phys. Rev. C 85, 034002 (2012).
  24. B. Long and C. J. Yang, Short-range nuclear forces in singlet channels, Phys. Rev. C 86, 024001 (2012).
  25. S. Wu and B. Long, Perturbative NN scattering in chiral effective field theory, Phys. Rev. C 99, 024003 (2019).
  26. M. P. Valderrama and D. R. Phillips, Power counting of contact-range currents in effective field theory, Phys. Rev. Lett. 114, 082502 (2015).
  27. V. D. Efros, W. Leidemann, and G. Orlandini, Response functions from integral transforms with a Lorentz kernel, Phys. Lett. B 338, 130 (1994).
  28. V. D. Efros, W. Leidemann, G. Orlandini, and N. Barnea, The Lorentz integral transform (LIT) method and its applications to perturbation-induced reactions, J. Phys. G: Nucl. Part. Phys. 34, R459 (2007).
  29. F. Marino, F. Bonaiti, S. Bacca, G. Hagen, and G. R. Jansen, Structure and dynamics of open-shell nuclei from spherical coupled-cluster theory, Phys. Rev. C 112, 014315 (2025).
  30. H. W. Griesshammer, J. Liao, J. A. McGovern, A. Nogga, and D. R. Phillips, Compton scattering on He4 with nuclear one- and two-body densities, Eur. Phys. J. A 60, 132 (2024).
  31. T. de Forest Jr. and J. Walecka, Electron scattering and nuclear structure, Adv. Phys. 15, 1 (1966).
  32. J. Carlson and R. Schiavilla, Structure and dynamics of few-nucleon systems, Rev. Mod. Phys. 70, 743 (1998).
  33. S. Bacca and S. Pastore, Electromagnetic reactions on light nuclei, J. Phys. G: Nucl. Part. Phys. 41, 123002 (2014).
  34. S. Martinelli, H. Kamada, G. Orlandini, and W. Glöckle, Longitudinal response functions of He3 and H3 by Lorentz kernel transformations, Phys. Rev. C 52, 1778 (1995).
  35. S. Bacca, Study of electromagnetic reactions on light nuclei with the lorentz integral transform method, Ph.D. Thesis, Johannes Gutenberg-Universität Mainz, Mainz, 2005.
  36. V. D. Efros, W. Leidemann, and G. Orlandini, Electron scattering response functions from their Stieltjes transforms, Few-Body Syst. 14, 151 (1993).
  37. E. Epelbaum, W. Glöckle, and U.-G. Meißner, Nuclear forces from chiral Lagrangians using the method of unitary transformation II: The two-nucleon system, Nucl. Phys. A 671, 295 (2000).
  38. X. Kong and F. Ravndal, Coulomb effects in low energy proton–proton scattering, Nucl. Phys. A 665, 137 (2000).
  39. D. B. Kaplan, Convergence of nuclear effective field theory with perturbative pions, Phys. Rev. C 102, 034004 (2020).
  40. W. Frank, D. J. Land, and R. M. Spector, Singular potentials, Rev. Mod. Phys. 43, 36 (1971).
  41. The nn-online, http://nn-online.org.
  42. V. G. J. Stoks, R. A. M. Klomp, M. C. M. Rentmeester, and J. J. de Swart, Partial-wave analysis of all nucleon-nucleon scattering data below 350 MeV, Phys. Rev. C 48, 792 (1993).
  43. M. P. Valderrama, M. S. Sánchez, C. J. Yang, B. Long, J. Carbonell, and U. van Kolck, Power counting in peripheral partial waves: The singlet channels, Phys. Rev. C 95, 054001 (2017).
  44. A. M. Gasparyan and E. Epelbaum, “Renormalization-group-invariant effective field theory” for few-nucleon systems is cutoff dependent, Phys. Rev. C 107, 034001 (2023).
  45. R. Peng, B. Long, and F.-R. Xu, Contact operators in renormalization of attractive singular potentials, Phys. Rev. C 110, 054001 (2024).
  46. C. J. Yang, Further theoretical study on the renormalization group aspect of perturbative corrections, Phys. Rev. C 112, 014004 (2025).
  47. R. Peng, B. Long, and F.-R. Xu, Perturbative renormalization of chiral nuclear forces at subleading order in the S13−D13 channel, Phys. Rev. C 112, 064004 (2025).
  48. O. Thim, A. Ekström, and C. Forssén, Perturbative χEFT calculations of the deuteron and triton up to N2LO, Phys. Rev. C 112, 064008 (2025).
  49. M. Pavon Valderrama, Regulator constraints for the perturbative renormalizability of attractive triplets, Phys. Rev. C 112, 064009 (2025).
  50. M. Pavon Valderrama, Reexamining the perturbative renormalizability of coupled triplets, Phys. Rev. C 113, 014001 (2026).
  51. D. R. Phillips, Higher-order calculations of electron–deuteron scattering in nuclear effective theory, Phys. Lett. B 567, 12 (2003).
  52. P. J. Mohr, D. B. Newell, B. N. Taylor, and E. Tiesinga, CODATA recommended values of the fundamental physical constants: 2022, Rev. Mod. Phys. 97, 025002 (2025).
  53. H. Krebs, E. Epelbaum, and U.-G. Meißner, Nuclear electromagnetic currents to fourth order in chiral effective field theory, Few-Body Syst. 60, 31 (2019).
  54. R. Schiavilla and V. R. Pandharipande, Elastic e−d scattering data and the deuteron wave function, Phys. Rev. C 65, 064009 (2002).
  55. W. Glöckle, G. Hasberg, and A. R. Neghabian, Numerical treatment of few body equations in momentum space by the spline method, Z. Phys. A 305, 217 (1982).
  56. S. N. More, S. König, R. J. Furnstahl, and K. Hebeler, Deuteron electrodisintegration with unitarily evolved potentials, Phys. Rev. C 92, 064002 (2015).
  57. S. N. More, S. K. Bogner, and R. J. Furnstahl, Scale dependence of deuteron electrodisintegration, Phys. Rev. C 96, 054004 (2017).
  58. G. G. Simon, F. Borkowski, Ch. Schmitt, V. H. Walther, H. Arenhövel, and W. Fabian, Inelastic electromagnetic form factors of the deuteron and exchange current effects, Nucl. Phys. A 324, 277 (1979).
  59. W. Fabian and H. Arenhövel, Electrodisintegration of deuterium including nucleon detection in coincidence, Nucl. Phys. A 314, 253 (1979).
  60. H. Arenhövel, W. Leidemann, and E. L. Tomusiak, General survey of polarization observables in deuteron electrodisintegration, Eur. Phys. J. A 23, 147 (2005).
  61. S. Christlmeier and H. W. Griesshammer, Pion-less effective field theory on low-energy deuteron electrodisintegration, Phys. Rev. C 77, 064001 (2008).
  62. C. J. Yang and D. R. Phillips, The longitudinal response function of the deuteron in chiral effective field theory, Eur. Phys. J. A 49, 122 (2013).
  63. Y. P. Teng and H. W. Griesshammer, On two nucleons near unitarity with perturbative pions, Eur. Phys. J. A 61, 211 (2025).
  64. S. Lyu, L. Zuo, R. Peng, S. König, and B. Long, Simplification of chiral nuclear forces near the unitarity limit, arXiv:2511.12522.
  65. S. König, Energies and radii of light nuclei around unitarity, Eur. Phys. J. A 56, 113 (2020).
  66. R. Peng, S. Lyu, S. König, and B. Long, Constructing chiral effective field theory around unnatural leading-order interactions, Phys. Rev. C 105, 054002 (2022).
  67. S. Bacca, N. Barnea, W. Leidemann, and G. Orlandini, Isoscalar monopole resonance of the alpha particle: A prism to nuclear Hamiltonians, Phys. Rev. Lett. 110, 042503 (2013).
  68. S. Kegel et al., Measurement of the α-particle monopole transition form factor challenges theory: A low-energy puzzle for nuclear forces?, Phys. Rev. Lett. 130, 152502 (2023).
  69. N. Michel, W. Nazarewicz, and M. Płoszajczak, Description of the proton-decaying 02+ resonance of the α particle, Phys. Rev. Lett. 131, 242502 (2023); 133, 239901(E) (2024).
  70. U.-G. Meißner, S. Shen, S. Elhatisari, and D. Lee, Ab initio calculation of the alpha-particle monopole transition form factor, Phys. Rev. Lett. 132, 062501 (2024).
  71. P. Yin, A. M. Shirokov, H. Li, B. Zhou, X. Zhao, S. Bacca, and J. P. Vary, α-particle monopole form factors within the ab initio no-core shell model, Phys. Rev. C 112, L031303 (2025).
  72. A. J. Andis, B. Lyu, S. Long, and S. König, Data: Perturbative eft calculation of the deuteron longitudinal response function, Zenodo, 2026, doi: 10.5281/zenodo.18039443.
  73. W. Glöckle, The Quantum Mechanical Few-body Problem (Springer, Berlin, 1983).
  74. D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskii, Quantum Theory of Angular Momentum: Irreducible Tensors, Spherical Harmonics, Vector Coupling Coefficients, 3nj Symbols (World Scientific, Singapore, 1988).
  75. M. Hjorth-Jensen, Applied data analysis and machine learning (2023), https://github.com/CompPhysics/MachineLearning.

Outline

Information

Phys. Rev. C 114, 024004– Published 31 August, 2026
  • Received 24 December 2025
  • Revised 28 April 2026
  • Accepted 29 June 2026
Article part of CHORUS
Crossmark - Check for updates button

©2026 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation