Three-Bosons in a Finite 4D Euclidean Volume


The formulation of the Minkowskian and Euclidean homogeneous Faddeev-Bethe-Salpeter (FBS) equations for a three-boson system with a contact interaction in an infinite four-dimensional (4D) volume are reviewed, and the Euclidean one in a finite 4D volume is derived and discussed. In an Euclidean 4D box the periodic boundary conditions leading to quantized momentum turns the continuum homogeneous FBS equation into a discretized form, with solutions determining the spectrum running above the scattering threshold. A truncated set of the discretized FBS equation is solved for three-pions in the maximum isospin state, with scattering length, pion mass and 4D volume from recent lattice QCD calculations, and the results of the approximate FBS equation for the ground state mass are in qualitative agreement with the lattice result. The approximate dependence of the ground state mass on the pion-pion scattering length, pion mass and 4D volume is also suggested.

This is a preview of subscription content, access via your institution.


  1. 1.

    C. Gattringer, C. B. Lang., Quantum Chromodynamics on the Lattice: An Introductory Presentation, Lect. Notes Phys. 788 (Springer, Berlin Heidelberg 2010)

  2. 2.

    M. Lüscher, Commun. Math. Phys. 105, 153 (1986)

    ADS  Article  Google Scholar 

  3. 3.

    M. Lüscher, Nucl. Phys. B 354, 531 (1991)

    ADS  Article  Google Scholar 

  4. 4.

    S.R. Beane, W. Detmold, T.C. Luu, K. Orginos, M.J. Savage, A. Torok, Phys. Rev. Lett. 100, 082004 (2008)

    ADS  Article  Google Scholar 

  5. 5.

    W. Detmold, M.J. Savage, A. Torok, S.R. Beane, T.C. Luu, K. Orginos, A. Parreño, Phys. Rev. D 78, 014507 (2008)

    ADS  Article  Google Scholar 

  6. 6.

    M. Mai, M. Döring, Phys. Rev. Lett. 122, 062503 (2019)

    ADS  Article  Google Scholar 

  7. 7.

    B. Hörz, A. Hanlon, Phys. Rev. Lett. 123, 142002 (2019)

    ADS  Article  Google Scholar 

  8. 8.

    T.D. Blanton, F. Romero-López, S.R. Sharpe, Phys. Rev. Lett. 124, 032001 (2020)

    ADS  Article  Google Scholar 

  9. 9.

    T. Frederico, Phys. Lett. B 282, 409 (1992)

    ADS  Article  Google Scholar 

  10. 10.

    J. Carbonell, V.A. Karmanov, Phys. Rev. C 67, 037001 (2003)

    ADS  Article  Google Scholar 

  11. 11.

    J. Carbonell, V.A. Karmanov, Few Body Syst. 49, 205 (2011)

    ADS  Article  Google Scholar 

  12. 12.

    L.H. Thomas, Phys. Rev. 47, 903 (1935)

    ADS  Article  Google Scholar 

  13. 13.

    V. Efimov, Phys. Lett. B 33, 563 (1970)

    ADS  Article  Google Scholar 

  14. 14.

    V. Efimov, Nucl. Phys. A 362, 45 (1981)

    ADS  Article  Google Scholar 

  15. 15.

    E. Ydrefors, J.H. Alvarenga Nogueira, V. Gigante, T. Frederico, V.A. Karmanov, Phys. Lett. B 770, 131 (2017)

    ADS  Article  Google Scholar 

  16. 16.

    V. Karmanov, P. Maris, Few Body Syst. 46, 95 (2009)

    ADS  Article  Google Scholar 

  17. 17.

    E. Ydrefors, J.H. Alvarenga Nogueira, V.A. Karmanov, T. Frederico, Phys. Lett. B 791, 276 (2019)

    ADS  MathSciNet  Article  Google Scholar 

  18. 18.

    E. Ydrefors, J.H.A. Nogueira, V.A. Karmanov, T. Frederico, Phys. Rev. D 101, 096018 (2020)

    ADS  MathSciNet  Article  Google Scholar 

  19. 19.

    G.V. Skorniakov, K.A. Ter-Martirosyan, Sov. Phys. JETP 4, 648 (1957)

    Google Scholar 

  20. 20.

    E. Nielsen, D.V. Fedorov, A.S. Jensen, E. Garrido, Phys. Rep. 347, 373 (2001)

    ADS  MathSciNet  Article  Google Scholar 

  21. 21.

    E.F. Suisso, J.P.B.C. de Melo, T. Frederico, Phys. Rev. D 65, 094009 (2002)

    ADS  Article  Google Scholar 

Download references


This work was partially supported by Fundação de Amparo à Pesquisa do Estado de São Paulo [2017/05660-0 (TF), 2016/25143-7 (EY)], Conselho Nacional de Desenvolvimento Científico e Tecnológico [308486/2015-3 (TF) and Project INCT-FNA 464898/2014-5].

Author information



Corresponding author

Correspondence to T. Frederico.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Frederico, T., Ydrefors, E. Three-Bosons in a Finite 4D Euclidean Volume. Few-Body Syst 62, 8 (2021).

Download citation