Il Nuovo Cimento A (1965-1970)

, Volume 42, Issue 3, pp 493–515 | Cite as

The discontinuity of a general feynman integral and regge poles in perturbation theory

  • B. Hamprecht


A formula for the complete discontinuity across the cut of an arbitrary Feynman integral is derived. One-variable dispersion relations for a certain range of the other variable are established. Mellin transforms are extended to Feynman integrals with indefinite coefficients ofs andt in the Feynman denominator. This extension is nontrivial. Finally a method is derived that allows treatment of the leading asymptotic behaviour in terms of Legendre functions instead of only in terms of powers of the variable going to infinity. It is shown that in all cases in which a leading\(( - t)^{\alpha _i (s)} \) behaviour has been obtained for a class of Feynman integrals summed, a Regge pole can explicitely be shown to exist in the Regge plane. The pole structure in the Regge plane is shown to be simpler than the one in the α-plane.


Regge Pole Pole Structure Landau Equation Feynman Integral Ladder Diagram 
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Si deduce una formula per la discontinuità completa nel taglio di un integrale di Feynman arbitrario. Si scrivono le relazioni di dispersione di una variabile per un certo intervallo di valori dell'altra variabile. Si estendono le trasformazioni di Mellin agli integrali di Feynman con coefficienti indefiniti dis et nel denominatore di Feynman. Questa estensione non è banale. Infine si deduce un metodo che permette di trattare il comportamento asintotico dominante in termini delle funzioni di Legendre invece che soltanto in termini di potenze della variabile tendente all'infinito. In tutti i casi in cui si è ottenuto un comportamento dominante\(( - t)^{\alpha _i (s)} \) per una classe di integrali di Feynman sommati, si può dimostrare esplicitamente che esiste un polo di Regge nel piano di Regge. Si dimostra che la struttura del polo nel piano di Regge è più semplice di quella nel piano α.


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  1. (1).
    J. S. R. Chisholm:Proc. Camb. Phil. Soc.,48, 300 (1952).ADSMathSciNetCrossRefGoogle Scholar
  2. (2).
    R. J. Eden: Brandeis Lecture Notes (1961).Google Scholar
  3. (3).
    T. T. Wu:Phys. Rev.,123, 678 (1961).ADSCrossRefGoogle Scholar
  4. (4).
    G. Tiktopoulos:Phys. Rev.,131, 480, 2373 (1963).ADSMathSciNetCrossRefGoogle Scholar
  5. (5).
    S. Mandelstam:Phys. Rev.,112, 1377 (1958).MathSciNetCrossRefGoogle Scholar
  6. (6).
    R. Omnès andM. Froissart:Mandelstam Theory and Regge Poles (New York, 1963).Google Scholar
  7. (7).
    S. C. Frautschi:Regge Poles and S-Matrix Theory (New York, 1963).Google Scholar
  8. (8).
    J. D. Bjorken andT. T. Wu:Phys. Rev.,130, 2566 (1963).ADSMathSciNetCrossRefGoogle Scholar
  9. (9).
    J. C. Polkinghorne:Journ. Math. Phys.,5, 431 (1963).ADSMathSciNetCrossRefGoogle Scholar
  10. (10).
    S. Mandelstam:Ann. of Phys.,19, 254 (1962).ADSMathSciNetCrossRefGoogle Scholar
  11. (11).
    A. R. Swift: Cambridge Preprint, to be published inNuovo Cimento.Google Scholar
  12. (12).
    P. G. Federbush andM. T. Grisaru:Ann. of Phys.,22, 263, 299 (1963).ADSMathSciNetCrossRefGoogle Scholar
  13. (13).
    I. G. Halliday:Nuovo Cimento,30, 177 (1963).MathSciNetCrossRefGoogle Scholar
  14. (14).
    L. D. Landau:Nucl. Phys.,13, 181 (1959).CrossRefGoogle Scholar
  15. (15).
    The usual formalism does not apply here, becausef may be indefinite.Google Scholar
  16. (16).
    J. C. Polkinghorne:Journ. Math. Phys.,4, 503 (1963).ADSMathSciNetCrossRefGoogle Scholar
  17. (17).
    E. T. Whittacker andG. N. Watson:A Course of Modern Analysis (Cambridge, 1962).Google Scholar
  18. (18).
    The rungs of the ladder are labelled byx 1,…x r, the other lines byy 1,…y k.Google Scholar
  19. (19).
    fg means hereD r(e f)=D r(e g).Google Scholar
  20. (20).
    Note thatk=2r−2.Google Scholar

Copyright information

© Società Italiana di Fisica 1966

Authors and Affiliations

  • B. Hamprecht
    • 1
  1. 1.Department of Applied Mathematics and Theoretical PhysicsUniversity of CambridgeCambridge

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