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Il Nuovo Cimento A (1965-1970)

, Volume 16, Issue 2, pp 345–375 | Cite as

Sum rules for Khuri amplitudes and light-cone Wilson expansion

  • M. Boiti
  • G. Immirzi
  • G. Maiella
  • F. Pempinelli
Article

Summary

The convolution theorem is used to obtain general sum rules for the Khuri amplitudes of the current correlation function (familiar from current algebra). The sum rules can be continued analytically in the Khuri λ-plane to negative integer values of λ; at these points the sum rules relate the finite-energy integral to the Fourier transform of distributions that, by the Gel'fand-Shilov method of analytic continuation, can be defined in a natural way as the current commutator and its derivatives on the light-cone. Assuming that the current commutator involved has a light-cone Wilson expansion, one can write the previously mentioned sum rules in terms of scaling amplitudes. Finally one considers the peculiar features of the sum rules obtained in the case of conserved currents.

Правила сумм для амплитуд Хури и разложение Вильсона на световом конусе

Резюме

Используется теорема свертки для получения общих правил сумм для амплитуд Хури корреляционной функции токов (известных из алгебры токов). Правила сумм могут быть продолжены аналитически в λ-плоскость Хури для отрицательных целых значений λ; в этих точках правила сумм связывают интеграл при конечных энергиях с фурье-преобразованием распределений, которые, с помощью метода аналитического продолжения Гельфанда-Шилова, могут быть определены естественным образом как коммутатор токов и его производные на световом конусе. Предполагая, что рассматриваемый коммутатор токов имеет вильсоновское разложение на световом конусе, можно записать вышеупомянутые правила сумм через масштабные амплитуды. В заключение рассматриваются характерные особенности правил сумм, полученных в случае сохраняющихся токов.

Riassunto

Si usa il teorema di convoluzione per ottenere regole di somma generali per le ampiezze di Khuri della funzione di correlazione fra correnti (familiare dall'algebra delle correnti). Le regole di somma possono essere continuate analiticamente, nel piano λ di Khuri, a valori interi negativi di λ; in questi punti le regole di somma collegano integrali ad energie finita con la trasformata di Fourier di distribuzioni che, in modo del tutto naturale, seguendo il metodo di prosecuzione analitica di Gel'fand-Shilov, possono essere definite come il commutatore di correnti e le sue derivate sul cono-luce. Assumendo poi che il commutatore di correnti considerato abbia uno sviluppo alla Wilson sul cono-luce, le regole di somma possono essere scritte in termini di ampiezze «scalate». Infine vengono considerate le peculiari caratteristiche delle regole di somma ottenute nel caso di correnti conservate.

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References

  1. (1).
    M. Boiti andF. Pempinelli:Lett. Nuovo Cimento,4, 781 (1972).CrossRefGoogle Scholar
  2. (2).
    S. Fubini andG. Furlan:Physics,1, 229 (1965).Google Scholar
  3. (3).
    G. Sartori andM. Tonin:Nuovo Cimento,54 A, 55 (1968).CrossRefADSGoogle Scholar
  4. (4).
    K. Bardakci andG. Segré:Phys. Rev.,159, 1263 (1967);J. Jersak andJ. Stern:Nuovo Cimento,59 A, 315 (1969);H. Leutwyler:Proceedings of the Summer School for Theoretical Physics, Karlsruhe, Springer Tracts in Modern Physics, Vol.50 (1969), p. 29;G. Furlan, R. Jengo andC. Rebbi:Nuovo Cimento,9 A, 487 (1972).CrossRefADSGoogle Scholar
  5. (5).
    K. W. Kendall:Proceedings of the 1971 International Symposium on Electron and Photon Interactions at High Energies (Ithaca, 1971).Google Scholar
  6. (6).
    B. L. Ioffe:Phys. Lett.,30 B, 123 (1969);R. A. Brandt:Phys. Rev. D,1, 2808 (1970).CrossRefADSGoogle Scholar
  7. (7).
    J. D. Bjorken:Phys. Rev.,179, 1547 (1969).CrossRefADSGoogle Scholar
  8. (8).
    H. Leutwyler andJ. Stern:Nucl. Phys.,20 B, 77 (1970);R. Jackiw, R. Van Royen andG. B. West:Phys. Rev. D,2, 2473 (1970).MathSciNetCrossRefADSGoogle Scholar
  9. (9).
    K. G. Wilson:Phys. Rev.,179, 1499 (1969);R. A. Brandt andG. Preparata:Nucl. Phys.,27 B, 541 (1971);W. Zimmermann:Lecture Notes in Physics Vol.17 (Berlin. 1973).MathSciNetCrossRefADSGoogle Scholar
  10. (10).
    N. Khuri:Phys. Rev.,132, 914 (1963);D. S. Freedman andJ. M. Wang:Phys. Rev.,153, 1596 (1967).MathSciNetCrossRefADSGoogle Scholar
  11. (11).
    R. Dashen andM. Gell-Mann:Phys. Rev. Lett.,17, 340 (1966);S. Fubini:Nuovo Cimento,43 A, 475 (1966).MathSciNetCrossRefADSGoogle Scholar
  12. (12).
    J. M. Cornwall, J. O. Corrigan andR. E. Norton:Phys. Rev. Lett.,24, 1141 (1970);Phys. Rev. D,3, 536 (1971).CrossRefADSGoogle Scholar
  13. (13).
    I. M. Gelfand andG. E. Shilov:Generalized Functions, Vol.1, Chap. 2, Sect.2 (New York, 1964).Google Scholar
  14. (15).
    G.S.: Chapt. 1, Sect.5.Google Scholar
  15. (16).
    R. Dolen, D. Horn andC. Schmidt:Phys. Rev.,166, 1768 (1968).CrossRefADSGoogle Scholar
  16. (17).
    See for instance:R. Gatto andP. Menotti:Phys. Rev. D,5, 1493 (1972).CrossRefADSGoogle Scholar
  17. (18).
    The relevance of this fact has been pointed out by:H. Leutwyler andP. Otterson:Theoretical problems in deep inelastic scattering, Talk presented at theFrascati Meeting, May 1972.Google Scholar
  18. (19).
    G. Preparata:Lectures in Light Cone Physics, Seventh Finnish Summer School in High-Energy Physics, June 1972.Google Scholar
  19. (20).
    R. Gatto andP. Menotti:Phys. Rev. D,5, 1493 (1972).CrossRefADSGoogle Scholar
  20. (22).
    J. W. Meyer andH. Suura:Phys. Rev.,160, 1366 (1967).CrossRefADSGoogle Scholar
  21. (23).
    See for instance:I. Todorov:Lecture Notes in Physics, Vol.17 (Berlin, 1973).Google Scholar
  22. (24).
    R. A. Brandt andG. Preparata:Nucl. Phys.,27 B, 541 (1971);C. G. Callan:Rendiconti S.I.F., Course LIV (New York, 1971).CrossRefADSGoogle Scholar
  23. (27).
    See ref. (8) andS. Ciccariello, R. Gatto, G. Sartori andM. Tonin:Ann. of Phys. 65, 265 (1971);M. Boiti, E. Donini andF. Pempinelli:Nuovo Cimento,3 A, 82 (1971).MathSciNetCrossRefADSGoogle Scholar
  24. (28).
    See ref. (5)K. W. Kendall:Proceedings of the 1971 International Symposium on Electron and Photon Interactions at High Energies (Ithaca, 1971). andM. Boiti, E. Donini andF. Pempinelli:Lett. Nuovo Cimento 2, 686 (1971);J. M., Cornwall andR. Jackiw:Phys. Rev. D,4, 367 (1971);H. Fritzsch andM. Gell-Mann:Tracts in Mathematics and Natural Sciences, Vol.2 (New York, 1971).Google Scholar

Copyright information

© Società Italiana di Fisica 1973

Authors and Affiliations

  • M. Boiti
    • 1
  • G. Immirzi
    • 1
  • G. Maiella
    • 1
  • F. Pempinelli
    • 1
  1. 1.Istituto di Fisica dell'UniversitàLecce

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