Bethe-Salpeter equations forq\(\bar q\) andqqq systems in the instantaneous approximation

  • A. N. Mitra
Article
Received:

Abstract

Bethe-Salpeter (B.S.) equations are formulated in a general way for unequal massq−q andqqq systems with pairwiseq−q andq1q2 inter-actions of the single-gluon exchange (QCD) and long-range (confining) types with a common colour (~λ(1)·λ(2)) and spin (~γ μ (1) γ μ (2) ) dependence for both. Spin reductions of these equations are achieved in a four dimensionally convariant manner by the method of Gordon decomposition which exhibits the structure of the spin-spin, spin-orbit and tensor terms in an elegant and compact fashion for any pairwise interaction, in preference to the usual procedure of reduction, to large and small components. The instantaneous approximation (IA), with its standard definition for a two-body system, and an extended one for the three-body system through a matching ansatz for the off-shell energy of the spectator particleq3 when a givenq1q2 pair is in interaction, is then used for a reduction of these B.S. equations to the threedimensional level. The latter equations which are written down in closely analogous forms forq−q andqqq systems for the general case of unequal mass kinematics, represent the main results of this paper, and are capable of straightforward extension to theqq−q−q system as well.

Keywords

Particle Acceleration Standard Definition Small Component Analogous Form Pairwise Interaction 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    E.E. Salpeter, H.A. Bethe: Phys. Rev.84, 1232 (1951)Google Scholar
  2. 2.
    M. Gell-Mann, F.E. Low: Phys. Rev.84, 350 (1951)Google Scholar
  3. 3.
    E.g., R. Karplus, A. Klein: Phys. Rev.87, 848 (1952)Google Scholar
  4. 4.
    E. Eichten et al.: Phys. Rev. Lett.34, 369 (1975)Google Scholar
  5. 5.
    A. de Rujula et al.: Phys. Rev.D12, 147 (1975)Google Scholar
  6. 6.
    D. Gromes: Nucl. Phys.B131, 80 (1977)Google Scholar
  7. 7.
    R. Barbieri et al.: Nucl. Phys.B105, 125 (1976)Google Scholar
  8. 8.
    A. Billoire, A. Morel: Nucl. Phys.B135, 131 (1978)Google Scholar
  9. 9.
    H.A. Bethe, E.E. Salpeter: Handbuch der Physik, Vol. 35, p. 88. Berlin, Heidelberg, New York: Springer 1967Google Scholar
  10. 10a.
    A. Henriques et al. Phys. Lett.64B, 85 (1976)Google Scholar
  11. 10b.
    W. Celmaster, F.S. Heynyey: Phys. Lett.D17, 3268 (1978)Google Scholar
  12. 11.
    For extensive reference to c−c spectroscopy, see e.g. A. W. Hendry, D.B. Lichtenberg: Rep. Prog. Phys.41, 1707 (1978)Google Scholar
  13. 12.
    N. Isgur, G. Karl: Phys. Rev.D18, 4187 (1978); ibid Phys. Rev.D19, 2653 (1979) D. Gromes: Nucl. Phys.B112, 213 (1976)Google Scholar
  14. 13a.
    L.J. Reinders: J. Phys.G4, 1241 (1978)Google Scholar
  15. 13b.
    U. Eilwanger: Nucl. Phys.B139, 422 (1978)Google Scholar
  16. 14.
    R.H. Dalitz: In Proc. XIII Intl. Conf. on HEP, Berkeley, 1966; Berkeley, Los Angeles: Univ. of California Press 1977Google Scholar
  17. 15.
    R.K. Bhaduri et al.: Phys. Rev. Lett.44, 1369 (1980)Google Scholar
  18. 16.
    Some early papers on the subject are: P. Narayanaswamy A. Pagnamenta; Nuovo Cimento53A, 635 (1968)Google Scholar
  19. 16.ii)
    C.H. L-Smith: Ann. Phys. (N.Y.)53, 521 (1969)Google Scholar
  20. 16.iii)
    M. Sundaresan, P. Watson: Ann. Phys. (N.Y.)59, 375 (1970)Google Scholar
  21. 17.
    R.P. Feynman, M. Kislinger, R. Ravindal: Phys. Rev.D, 3, 2706 (1971); referred to as FKRGoogle Scholar
  22. 18.
    M. Böhm, H. Joos M. Krammer: Acta Phys. Austriaca, Suppl.XI, 3 (1973), and a large number of subsequent papersGoogle Scholar
  23. 19.
    For a review of Wick rotation, etc., see, e.g., N. Nakanishi: Prog. Theo. Phys. Suppl.43, 1 (1969)Google Scholar
  24. 20.
    M. Böhm, M. Krammer: Nucl. Phys.B120, 113 (1977); and for earlier references on the subject; see also, D. Flamm, F. Schoberl: Univ of Vienna 1979; to be publishedGoogle Scholar
  25. 21.
    R.F. Meyer: Nucl. Phys.B71, 226 (1974)Google Scholar
  26. 22.
    C. Alibaso, G. Schierholtz: Nucl. Phys.B126, 461 (1977); see also, J.R. Henley: Phys. Rev.D20, 2532 (1979)Google Scholar
  27. 23.
    H. Leutwyler, J. Stern: Ann. Phys. (N.Y.)112, 94 (1978)Google Scholar
  28. 24.
    H. Leutwyler, J. Stern: Phys. Lett.73B, 75 (1978)Google Scholar
  29. 25.a
    M. Levy: Phys. Rev.88, 72 (1952)Google Scholar
  30. 25.b
    A. Klein: H. Phys. Rev.90, 1101 (1963)Google Scholar
  31. 26.
    A.N. Mitra, I. Santhanam Z. Phys. C8, 33 (1981)Google Scholar
  32. 27.
    J.D. Bjorken, S.D. Drell: Relativistic quantum Mechanics New York: McGraw-Hill 1964Google Scholar
  33. 28.
    A.N. Mitra: Phys. Lett.89B, 65 (1979)Google Scholar

Copyright information

© Springer-Verlag 1981

Authors and Affiliations

  • A. N. Mitra
    • 1
  1. 1.Department of PhysicsUniversity of DelhiDelhiIndia

Personalised recommendations