Few-Body Systems

, 59:27 | Cite as

Equation for the Nakanishi Weight Function Using the Inverse Stieltjes Transform

  • V. A. Karmanov
  • J. Carbonell
  • T. Frederico
Part of the following topical collections:
  1. Light Cone 2017


The bound state Bethe–Salpeter amplitude was expressed by Nakanishi in terms of a smooth weight function g. By using the generalized Stieltjes transform, we derive an integral equation for the Nakanishi function g for a bound state case. It has the standard form \(g= \hat{\mathcal V} g\), where \(\hat{\mathcal V} \) is a two-dimensional integral operator. The prescription for obtaining the kernel \({\mathcal V} \) starting with the kernel K of the Bethe–Salpeter equation is given.



We are indebted to G. Salmé for useful discussions. T.F. thanks CNPq, CAPES and FAPESP of Brazil. V.A.K. thanks the support of FAPESP, the Grant #2015/22701-6 and is sincerely grateful for kind hospitality to Theoretical Nuclear Physics Group in ITA, São José dos Campos, Brazil, where the main part of this research was carried out.


  1. 1.
    E.E. Salpeter, H.A. Bethe, A relativistic equation for bound-state problems. Phys. Rev. 84, 1232 (1951)MathSciNetCrossRefzbMATHADSGoogle Scholar
  2. 2.
    G.C. Wick, Properties of the Bethe–Salpeter wave functions. Phys. Rev. 96, 1124 (1954)MathSciNetCrossRefzbMATHADSGoogle Scholar
  3. 3.
    J. Carbonell, V.A. Karmanov, Solving Bethe–Salpeter scattering state equation in Minkowski space. Phys. Rev. D 90, 056002 (2014)CrossRefADSGoogle Scholar
  4. 4.
    K. Kusaka, A.G. Williams, Solving the Bethe–Salpeter equation for scalar theories in Minkowski space. Phys. Rev. D 51, 7026 (1995)CrossRefADSGoogle Scholar
  5. 5.
    K. Kusaka, K. Simpson, A.G. Williams, Solving the Bethe–Salpeter equation for bound states of scalar theories in Minkowski space. Phys. Rev. D 56, 5071 (1997)CrossRefzbMATHADSGoogle Scholar
  6. 6.
    V.A. Karmanov, J. Carbonell, Solving Bethe–Salpeter equation in Minkowski space. Eur. Phys. J. A 27, 1 (2006)CrossRefzbMATHADSGoogle Scholar
  7. 7.
    J. Carbonell, V.A. Karmanov, Cross-ladder effects in Bethe–Salpeter and light-front equations. Eur. Phys. J. A 27, 11 (2006)CrossRefADSGoogle Scholar
  8. 8.
    T. Frederico, G. Salmè, M. Viviani, Quantitative studies of the homogeneous Bethe–Salpeter equation in Minkowski space. Phys. Rev. D 89, 016010 (2014)CrossRefADSGoogle Scholar
  9. 9.
    T. Frederico, G. Salmè, M. Viviani, Two-body scattering states in Minkowski space and the Nakanishi integral representation onto the null plane. Phys. Rev. D 85, 036009 (2012)CrossRefADSGoogle Scholar
  10. 10.
    T. Frederico, G. Salmè, M. Viviani, Solving the inhomogeneous Bethe–Salpeter equation in Minkowski space: the zero-energy limit. Eur. Phys. J. C 75, 398 (2015)CrossRefADSGoogle Scholar
  11. 11.
    W. de Paula, T. Frederico, G. Salmè, M. Viviani, Advances in solving the two-fermion homogeneous Bethe–Salpeter equation in Minkowski space. Phys. Rev. D 94, 071901 (2016)CrossRefADSGoogle Scholar
  12. 12.
    C. Gutierrez, V. Gigante, T. Frederico, G. Salmè, M. Viviani, L. Tomio, Bethe–Salpeter bound-state structure in Minkowski space. Phys. Lett. B 759, 131 (2016)CrossRefzbMATHADSGoogle Scholar
  13. 13.
    T. Frederico, J. Carbonell, V. Gigante, V.A. Karmanov, Inverting the Nakanishi integral relation for a bound state Euclidean Bethe–Salpeter amplitude. Few-Body Syst. 56, 549 (2016)CrossRefADSGoogle Scholar
  14. 14.
    T. Frederico, J. Carbonell, V.A. Karmanov, Euclidean to Minkowski Bethe–Salpeter amplitude and observables. Eur. Phys. J. C 77, 58 (2017)CrossRefADSGoogle Scholar
  15. 15.
    N. Nakanishi, Partial-wave Bethe–Salpeter equation. Phys. Rev. 130, 1230 (1963)MathSciNetCrossRefADSGoogle Scholar
  16. 16.
    N. Nakanishi, General survey of the theory of the Bethe–Salpeter equation. Prog. Theor. Phys. Suppl. 43, 1 (1969)MathSciNetCrossRefzbMATHADSGoogle Scholar
  17. 17.
    N. Nakanishi, Graph Theory and Feynman Integrals (Gordon and Breach, New York, 1971)zbMATHGoogle Scholar
  18. 18.
    J. Carbonell, V.A. Karmanov, M. Mangin-Brinet, Electromagnetic form factor via Bethe–Salpeter amplitude in Minkowski space. Eur. Phys. J. A 39, 53 (2009)CrossRefADSGoogle Scholar
  19. 19.
    R.E. Cutkosky, Solutions of the Bethe–Salpeter equation. Phys. Rev. 96, 1135 (1954)MathSciNetCrossRefzbMATHADSGoogle Scholar
  20. 20.
    V. Gigante, J.H. Alvarenga Nogueira, E. Ydrefors, C. Gutierrez, V.A. Karmanov, T. Frederico, Bound state structure and electromagnetic form factor beyond the ladder approximation. Phys. Rev. D 95, 056012 (2017)CrossRefADSGoogle Scholar
  21. 21.
    J. Carbonell, T. Frederico, V.A. Karmanov, Bound state equation for the Nakanishi weight function. Phys. Lett. B 769, 418 (2017)CrossRefzbMATHADSGoogle Scholar
  22. 22.
    J.H. Schwarz, The generalized Stieltjes transform and its inverse. J Math. Phys. 46, 014501 (2005). arXiv:math-ph/0405050v1 MathSciNetCrossRefGoogle Scholar
  23. 23.
    D.B. Sumner, An inversion formula for the generalized Stieltjes transform. Bull. Am. Math. Soc. 55, 174 (1949)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    G. Salmè, Private communication, November 2016Google Scholar
  25. 25.
    V.D. Efros, Calculation of inclusive transition spectra and the reaction cross sections without wave functions. Sov. J. Nucl. Phys. 41, 949 (1985)Google Scholar
  26. 26.
    V.D. Efros, W. Leidemann, G. Orlandini, N. Barnea, The Lorentz integral transform (LIT) method and its applications to perturbation-induced reactions. J. Phys. G Nucl. Part. Phys. 34, R459 (2007)CrossRefADSGoogle Scholar
  27. 27.
    G. Orlandini, F. Turro, Integral transform methods: a critical review of various kernels. Few Body Syst. 58, 76 (2017)CrossRefADSGoogle Scholar

Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Lebedev Physical InstituteMoscowRussia
  2. 2.Institut de Physique Nucleaire, IN2P3-CNRSUniversité Paris-SudOrsay CedexFrance
  3. 3.Instituto Tecnológico de AeronáuticaDCTAS. José dos CamposBrazil

Personalised recommendations