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Il Nuovo Cimento A (1971-1996)

, Volume 43, Issue 2, pp 258–324 | Cite as

Practical theory of three-particle states

II. — Relativistic, spin zero
  • D. Z. Freedman
  • C. Lovelace
  • J. M. Namyslowski
Article

Summary

We obtain covariant equations for the scattering of composite particles. They are coupled linear integral equations in one variable. The solutions satisfy three-particle unitarity, and all observables in three-particle systems can be expressed in terms of them. The equations are derived from field theory, the basic approximation being that each two-particle subsystem is dominated by its bound states and resonances. However, the final equations involve only the wave functions of the composite particles, and not the original Lagrangian. Overlapping resonances are correctly taken into account, and some three-body forces are also included. The « potential » is essentially the Peierls mechanism, and its imaginary part gives the interference effect between overlapping resonances. Our equations are different from and simpler than those of Alessandrini and Omnès, because we eliminate the relative energies in a way compatible with the Landau-Cutkosky rules. The present paper only gives the equations when the elementary particles are spinless (unequal masses).

Riassunto

Si ottengono equazioni covarianti per lo scattering di particelle composte. Esse sono equazioni integrali lineari accoppiate in una variabile. Le soluzioni soddisfano l’unitarietà di tre particelle e tutti gli osservabili nei sistemi di tre particelle possono essere espressi in funzione di esse. Le equazioni sono dedotte dalla teoria dei campi, con l’approssimazione fondamentale che ogni sottosistema di due particelle sia dominato dai suoi stati legati a risonanze. Tuttavia le equazioni finali implicano solo le funzioni d’onda delle particelle composte e non il lagrangiano originario. Si tiene correttamente conto delle risonanze sovrapposte e si includono anche alcune forze di tre corpi. Il « potenziale » è essenzialmente il meccanismo di Peierls e la sua parte immaginaria dà l’effetto di interferenza fra le risonanze sovrapposte. Le nostre equazioni sono diverse e più semplici di quelle diAlessandrini eOmnès, perchè si sono eliminate le energie relative in maniera compatibile con le regole di Landau-Cutkosky. Nel presente lavoro si danno le equazioni solo per il caso in cui le particelle elementari sono prive di spin (masse disuguali).

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References

  1. (2).
    L. D. Faddeev:Žurn. Ėksp. Teor. Fiz.,39, 1459 (1960).MathSciNetGoogle Scholar
  2. (3).
    A. N. Mitra:Nucl. Phys.,32, 529 (1962).CrossRefGoogle Scholar
  3. (7).
    J. Kirz, J. Schwartz andR. D. Tripp:Phys. Rev.,130, 2481 (1963).ADSCrossRefGoogle Scholar
  4. (8).
    G. C. Wick:Ann. of Phys.,18, 65 (1962).ADSMathSciNetCrossRefGoogle Scholar
  5. (10).
    Ya. A. Smorodinskii:Atomnaya Energiya,14, 110 (1963).Google Scholar
  6. (11).
    E. E. Salpeter andH. A. Bethe:Phys. Rev.,84, 1232 (1951).ADSMathSciNetCrossRefGoogle Scholar
  7. (13).
    R. Cutkosky:Phys. Rev. Lett.,4, 624 (1960).ADSCrossRefGoogle Scholar
  8. (14).
    J. G. Taylor:Suppl. Nuovo Cimento,1, 859 (1964).Google Scholar
  9. (15).
    R. Haag:Phys. Rev.,112, 669 (1958).ADSMathSciNetCrossRefGoogle Scholar
  10. (16).
    W. Zimmermann:Nuovo Cimento,10, 597 (1958).CrossRefGoogle Scholar
  11. (17).
    K. Nishijima:Progr. Theor. Phys.,10, 549 (1953).ADSMathSciNetCrossRefGoogle Scholar
  12. (18).
    H. Araki:Ann. of Phys.,11, 260 (1960).ADSMathSciNetCrossRefGoogle Scholar
  13. (20).
    H. Lehmann, K. Symanzik andW. Zimmermann:Nuovo Cimento,1, 205 (1955).MathSciNetCrossRefGoogle Scholar
  14. (23).
    R. J. Eden:Proc. Roy. Soc., A215, 133 (1952).ADSCrossRefGoogle Scholar
  15. (24).
    S. Weinberg:Phys. Rev.,130, 776 (1963).ADSMathSciNetCrossRefGoogle Scholar
  16. (25).
    R. Cirelli andG. Stabilini:Suppl. Nuovo Cimento,20, 157 (1961).MathSciNetCrossRefGoogle Scholar
  17. (27).
    G. Källén:Helv. Phys. Acta,25, 416 (1952).Google Scholar
  18. (28).
    H. Lehmann, K. Symanzik andW. Zimmermann:Nuovo Cimento,2, 425 (1955).MathSciNetCrossRefGoogle Scholar
  19. (29).
    O. I. Zav’yalov, M. K. Polivanov andS. S. Khoruzhii:Žurn. Ėksp. Teor. Fiz.,45, 1654 (1963).Google Scholar
  20. (30).
    J. Wright andM. Scadron:Nuovo Cimento,34, 1571 (1964).MathSciNetCrossRefGoogle Scholar
  21. (31).
    G. Wanders:Phys. Rev.,104, 1782 (1956).ADSMathSciNetCrossRefGoogle Scholar
  22. (32).
    G. F. Chew andS. Mandelstam:Phys. Rev.,119, 467 (1960).ADSMathSciNetCrossRefGoogle Scholar
  23. (33).
    D. Stoyanov andA. N. Tavkhelidze:Phys. Lett.,13, 76 (1964).ADSMathSciNetCrossRefGoogle Scholar
  24. (36).
    W. Zimmermann:Nuovo Cimento,13, 503 (1959).CrossRefGoogle Scholar
  25. (37).
    M. Baker andR. Blankenbecler:Phys. Rev.,128, 415 (1962).ADSMathSciNetCrossRefGoogle Scholar
  26. (38).
    C. Lovelace:Proc. Roy. Soc., A289, 547 (1966).ADSCrossRefGoogle Scholar
  27. (39).
    R. F. Peierls:Phys. Rev. Lett.,6, 641 (1961).ADSCrossRefGoogle Scholar
  28. (40).
    R. C. Hwa:Phys. Rev.,130, 2580 (1963).ADSMathSciNetCrossRefGoogle Scholar
  29. (41).
    M. Olsson andG. B. Yodh:Phys. Rev. Lett.,10, 353 (1963).ADSCrossRefGoogle Scholar
  30. (43).
    S. Bergia, F. Bonsignori andA. Stanghellini:Nuovo Cimento,15, 1073 (1960).CrossRefGoogle Scholar
  31. (44).
    S. W. MacDowell:Phys. Rev.,116, 774 (1959).ADSMathSciNetCrossRefGoogle Scholar
  32. (45).
    M. I. Shirokov:Žurn. Ėksp. Teor. Fiz.,40, 1387 (1961).Google Scholar
  33. (46).
    A. McKerrell:Nuovo Cimento,34, 1289 (1964).MathSciNetCrossRefGoogle Scholar
  34. (47).
    M. Jacob andG. C. Wick:Ann. of Phys.,7, 404 (1959).ADSMathSciNetCrossRefGoogle Scholar
  35. (48).
    Our kinematic factors differ from Wick’s because of the normalization of the two-particle states. His normalization unfortunately introduces a spurious kinematic singularity which eventually cancels. For this reason, care must be taken in applying the formulae of ref. (8) directly.ADSMathSciNetCrossRefGoogle Scholar
  36. (49).
    G. Murtaza:Nuovo Cimento,39, 195 (1965).ADSMathSciNetCrossRefGoogle Scholar
  37. (51).
    F. Coester:Helv. Phys. Acta,38, 7 (1965).MathSciNetGoogle Scholar

Copyright information

© Società Italiana di Fisica 1966

Authors and Affiliations

  • D. Z. Freedman
    • 1
    • 2
  • C. Lovelace
    • 1
    • 2
  • J. M. Namyslowski
    • 1
    • 2
  1. 1.Imperial CollegeLondon
  2. 2.CERNGeneva

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