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Il Nuovo Cimento A (1965-1970)

, Volume 49, Issue 4, pp 716–730 | Cite as

Theory of reggeized bootstraps.—III

  • W. J. Abbe
  • P. Nath
  • Y. N. Srivastava
Article

Summary

A theory for bootstrapping entire Regge eigentrajectories of a coupled-channel system is discussed. This is an extension of a recently proposed theory for bootstrapping Regge trajectories in the framework of single-channel processes. The formalism is based on a Regge representation for a partial-wave eigenamplitude which satisfies the following properties regardless of the number of eigentrajectories included; it has the correct threshold behavior for the real and the imaginary parts and it is unitary. The total scattering amplitudes are constructed in terms of eigentrajectories which have the correct Mandelstam-cut structure in thes-t-u plane and are also unitary. The representations are expected to converge rapidly in terms of the number of trajectories since the amplitudes used have been modified to take account of the infinite set ofJ-singularities in their Born limit. The relationship of Levinson's theorem to the motion of Regge eigenpoles is discussed. Invoking crossing, a set of bootstrap equations is obtained for a self-consistent determination of the Regge trajectories which occur as the sole input in the formalism.

Keywords

Regge Trajectory Bootstrap Equation Teorema Regge Zero Correct Threshold Behavior 
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.

Теория «бутстрзпа» для Редже траекторий.—III

Резюме

Обсуждается теория бутстрэпа для собственных Редже траекторий в двухканальной системе. Это представляет расширение недавно предложенной теории бутстрэп для Редже траекторий в рамках одноканальных процессов. Формализм основаан на Редже представлении для парциальной собственной амплитуды, которая удовлетворяет следующим свойствам, независимо от числа рассматриваемых собственных траекторий, амплитуда имеет правильное пороговое поведение для вещественной и мнимой частей, и амплитуда является унитарной. Суммарные амплитуды рассеяния конструируются в терминах собственных траекторий, которые имеют правильную структуру Мандельстама для разрезов вs-t-u плоскости и являются также унитарными. Ожидается, что представления быстро сходятся в терминах числа траекторий, т.к. использованные амплитуды видоизменены с учетом бесконечной системыJ-сингулярностей в Борновском пределе. Обсуждается свяэр теоремы Левинсона с движением собственных полюсов Редже. Применяя кроссинг, получается система бутстрэп-уравнений для самосогласованного определения траекторий Редже, которые оказываются единственным предположением в этом формализме.

Riassunto

Si discute una teoria per eseguire il bootstrap di intere autotraiettorie di Regge di un sistema di canali accoppiati. Si tratta dell'estensione di una teoria proposta di recente per eseguire il bootstrap delle traiettorie di Regge nello schema dei processi di un solo canale. Il formalismo si basa sulla rappresentazione di Regge di un'autoampiezza di onda parziale che soddisfa le seguenti proprietà a, prescindere dal numero di autotraiettorie incluse: essa ha il corretto comportamento di soglia per le parti reale ed imaginaria ed è unitaria. Si costruiscono le ampiezze di scattering totali in funzioni di autotriettorie che hanno la corretta struttura del taglio di Mandelstam nel pianos-t-u e sono anche unitarie. Ci si attende che le rappresentazioni convergano rapidamente in base al numero delle traiettorie poiché le ampiezze usate sono state modificate per tenere conto del gruppo infinito di singolaritàJ nel loro limite di Born. Si discute il rapporto fra il teorema di Levinson e il moto degli autopoli di Regge. Ricorrendo l'incrocio, si ottiene un gruppo di equazioni di bootstrap per una determinazione autocoerente delle traiettorie di Regge che si presentano come gli unici dati del formalismo.

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References

  1. (1).
    W. J. Abbe, P. Kaus, P. Nath andY. N. Srivastava:Phys. Rev.,141, B 1513 (1966). Hereafter called I.ADSCrossRefGoogle Scholar
  2. (2).
    W. J. Abbe, P. Kaus, P. Nath andY. N. Srivastava:Phys. Rev.,140, B 1595 (1965). Hereafter called II.ADSMathSciNetCrossRefGoogle Scholar
  3. (3).
    S. C. Frautschi, P. Kaus andF. Zachariasen:Phys. Rev.,133, B 1607 (1964). See alsoD. Hankins, P. Kaus andC. J. Pearson:Phys. Rev.,137, B 1034 (1965)ADSMathSciNetCrossRefGoogle Scholar
  4. (4).
    G. F. Chew:Phys. Rev.,129, 2363 (1963);G. F. Chew andC. E. Jones:Phys. Rev.,135, B 208 (1964);M. Kretzchmar:Nuovo Cimento,32, 1405 (1964);39, 835 (1965).ADSMathSciNetCrossRefGoogle Scholar
  5. (5).
    W. J. Abbe, P. Kaus, P. Nath andY. H. Srivastava:Bull. Am. Phys. Soc.,11, 381 (1966). See also,W. J. Abbe:Ph. D. Thesis (1965) University of California, Riverside (unpublished).Google Scholar
  6. (6).
    For a bibliography of such attempts seeLectures on Bootstraps byF. Zachariasen:Pacific International Summer School in Physics, August 1965,Honolulu, Hawaii.Google Scholar
  7. (8).
    The representation based on the asymptotic behavior (2.1) has been tested in a physical situation in the context of πN scattering and was found to be consistent with the available experimental data. SeeW. J. Abbe, P. Nath andY. N. Srivastava:Nuovo Cimento,45 A, 675 (1966).ADSCrossRefGoogle Scholar
  8. (9).
    Seee. g.,P. Kaus,P. Nath andY. N. Srivastava:Phys. Rev.,138, B 726 (1965).ADSMathSciNetCrossRefGoogle Scholar
  9. (10).
    For a discussion of this asymptotic form for the trajectories see ref. (1).ADSCrossRefGoogle Scholar
  10. (11).
    This general form is shared by relativistic trajectories as shown byV. N. Gribov andI. Ya. Pomeranchuk:Žurn. Ėksp. Teor. Fiz.,16, 220 (1963), under very general assumptions and independent of an underlying Hamiltonian.Google Scholar
  11. (12).
    Although the partial-waveS-matrix used here was derived assuming only simple poles in the λ-plane, it may be pointed out that the present procedure can be very easily generalized to include more complicated analytic structure in the λ-plane (e.g. branch cuts) if this is demanded by the relativistic problem.Google Scholar
  12. (13).
    Seee.g.,R. G. Newton:Journ. Math. Phys.,2, 188 (1961);P. Nath andY. N. Srivastava:Nuovo Cimento,138, 1192 (1965).ADSCrossRefGoogle Scholar
  13. (14).
    The argument can be extended to include alll-values using the Wong continuation, see ref. (2).Google Scholar
  14. (16).
    The present procedure is based on the method suggested to two of us (P. N. and Y. N. S.) by Profs.F. Calogero andA. Degasperis (private communication).Google Scholar
  15. (17).
    Our definition of Ricatti-Bessel functions follows that ofNuovo Cimento,27, 261 (1963).Google Scholar

Copyright information

© Società Italiana di Fisica 1967

Authors and Affiliations

  • W. J. Abbe
    • 1
  • P. Nath
    • 2
  • Y. N. Srivastava
    • 3
  1. 1.Department of PhysicsUniversity of GeorgiaAthens
  2. 2.Department of PhysicsUniversity of PittsburghPittsburgh
  3. 3.Department of PhysicsNortheastern UniversityBoston

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