Duality in Strong Interaction Physics

  • M. Jacob
Conference paper
Part of the Acta Physica Austriaca book series (FEWBODY, volume 6/1969)


The purpose of this series of lectures is not to introduce a theory, but rather an approach to strong interaction physics. This approach appears at present as fruitful and exciting and many papers exploring the consequences of duality have recently been published or are being circulated in preprint form. Nevertheless, I should mention to begin with that the whole matter is still in a controversial form. There is not yet agreement on how far duality can be used, since no precise definition for it is yet available. As a consequence, results presented here as definitive success are still bitterly challenged there. I will therefore not even try to start by a tentative definition which will emerge only as we proceed. In any case, after Horn’s series of lectures [1] on the use of finite energy sum rules, a very good idea of what it should be has already been gathered.


Partial Wave Regge Trajectory Partial Wave Amplitude Channel Pole Channel Resonance 
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References and Footnotes

  1. 1.
    D. Horn, Lecture Notes, Schladming Winter School 1969.Google Scholar
  2. 2.
    V. Barger, Communication at the Vienna Conference (1968).Google Scholar
  3. C. Schmid, CERN preprint TH.960, to be published in Nuovo Cimento.Google Scholar
  4. A. H. Rosenfeld, Tables, UCRL-8030 (1969).Google Scholar
  5. 3.
    Continuation to complex values of the angular momentum has to be made separately for even and odd J values since (−1)J does not fulfil the conditions of the Carlson theorem. The absence of an absorptive part in either the s or u channel, say, will therefore lead to two coincident t channel Regge trajectories with opposite signatures.Google Scholar
  6. 4.
    We will not consider here in detail the implication of the splitting of the A2 [5]. We will consider the A2 as the nonet partner of the f. If the two A2 peaks have to be associated with the same set of quantum numbers, we would then expect a similar splitting for all the nonet members.Google Scholar
  7. 5.
    W. Kienzle, Philadelphia Conference on Meson Spectroscopy (1968).Google Scholar
  8. H. Benz et al., Phys. Letters 28B, 233 (1968).ADSGoogle Scholar
  9. 6.
    Writing a once subtracted dispersion relation for α(t), one sees that the observed widths of meson resonances cannot support the large observed slope.Google Scholar
  10. P. D. B. Collins, R. C. Johnson and E. Squires, Phys. Letters 26B, 223 (1968).ADSGoogle Scholar
  11. 7.
    S. Mandelstam, Phys. Rev. 166, 1539 (1968).ADSCrossRefGoogle Scholar
  12. K. Dietz, Lecture Notes, Schladming Winterschool 1969.Google Scholar
  13. 8.
    B. E. Y. Svensson, CERN Report 67-24 (1967); L. van Hove, CERN Lecture Notes (1968).Google Scholar
  14. M. Jacob, Herceg Novi Lecture Notes (1968).Google Scholar
  15. 9.
    A sum of resonance contributions may be a good approximation to an inelastic (or charge exchange)amplitude. It is difficult to test in any definite way resonance saturation since there is much ambiguity at parametrizing resonances away from the poles, especially when they are highly inelastic.Google Scholar
  16. 10.
    G. F. Chew, Comments Nucl. Particle Phys. 1, 121 (1967).Google Scholar
  17. R. Dolen, D. Horn and C. Schmid, Phys. Rev. 166, 1768 (1968).ADSCrossRefGoogle Scholar
  18. C. Schmid, Phys. Rev. Letters 20, 628 (1968).ADSCrossRefGoogle Scholar
  19. G. F. Chew and A. Pignotti, Phys. Rev. Letters 20, 1078 (1968).ADSCrossRefGoogle Scholar
  20. 11.
    The imaginary part (say) of a Regge amplitude cannot reproduce the wiggles associated with the direct channel resonances. Nevertheless, we expect at best the Regge amplitude to represent what we would obtain smearing out all resonance peaks. It should correspond to a good average even for several moments, which defines a semilocal approximation.Google Scholar
  21. 12.
    Continuation to positive t values located in the double spectral function region is, of course, not possible from a phase shift or a multipole expansion. Nevertheless, it is done on the ground that the zero width (real a) approximation is not too far from the actual case to still yield interesting results.Google Scholar
  22. 13.
    C. Schmid, Phys. Rev. Letters 20, 689 (1968).ADSCrossRefGoogle Scholar
  23. P. D. B. Collins, R. C. Johnson and E. J. Squires, Phys. Letters 27B, 23 (1968).ADSGoogle Scholar
  24. C. Schmid, CERN Preprint TH.958, to be published in Nuovo Cimento.Google Scholar
  25. see also, Ref.[8], M. Jacob, Herceg Novi Lecture Notes (1968).Google Scholar
  26. 14.
    C. Lovelace, Proceedings of the Heidelberg Conference (1967).Google Scholar
  27. 15.
    C. B. Chiu and A. Kotanski, Nuclear Physics 7, 615 (1968); 8, 553 (1969).CrossRefGoogle Scholar
  28. 16.
    A second sheet pole, with dominant effect, is the simplest way to interpret the presence of an Argand loop. If further imposes the factorization property attached to a resonance. It is, however, not the only way of producing an Argand loop.Google Scholar
  29. 17.
    C. Lovelace, Contribution to the Vienna Conference (1968).Google Scholar
  30. 18.
    H. Harari, Phys. Rev. Letters 20, 1395 (1968); Contribution to the Vienna Conference (1968)ADSCrossRefGoogle Scholar
  31. P. G. O. Freund, Phys. Rev. Letters 20, 235 (1968).ADSCrossRefGoogle Scholar
  32. 19.
    D. Morrison, Phys. Letters 22, 528 (1966).ADSCrossRefGoogle Scholar
  33. 20.
    C. B. Chiu and J. Finkelstein, Phys. Letters 27B, 510 (1968).ADSGoogle Scholar
  34. 21.
    R. H. Dalitz, Les Houches Lecture Notes (1965).Google Scholar
  35. 22.
    The decoupling of the Σ and Y1*(1910) from the ̄KN channel imposes a particular F/D ratio α = F/F+D=½.Google Scholar
  36. 23.
    Standard absorption collections using the Baker-Blankenbecler formalism, say [24], will leave the amplitude real if the initial Regge amplitude is real. To the contrary, the phases of the spin flip and non-spin flip amplitude become different when originally complex.Google Scholar
  37. 24.
    M. Baker and Blankenbecler, Phys. Rev. 128, 415 (1962).MathSciNetADSCrossRefGoogle Scholar
  38. 25.
    L. van Hove, Phys. Letters 24B, 183 (1967).ADSGoogle Scholar
  39. L. Durand III, Phys. Rev. 161, 1610 (1967).ADSCrossRefGoogle Scholar
  40. 26.
    R. Jengo, Phys. Letters 28B, 261 (1968); 28B, 606 (1969).ADSGoogle Scholar
  41. 27.
    V. A. Alessandrini, D. Amati and E. J. Squires, Phys. Letters 27B, 463 (1968).ADSGoogle Scholar
  42. 28.
    V. Barger and M. Olsson, Phys. Rev. 146, 980 (1966).ADSCrossRefGoogle Scholar
  43. 29.
    H. Harari, Phys. Rev. Letters 22, 562 (1969).ADSCrossRefGoogle Scholar
  44. 30.
    J. Rosner, Tel Aviv University preprint (1968).Google Scholar
  45. G. P. Canning, University of Oxford preprint (1969).Google Scholar
  46. 31.
    With standard nonet mixing the φ(ω) and f′ (f) have only strange (non-strange) quarks. This leads to the “ideal” nonet mass formula where two states (I=1 and I=0) are degenerate in mass and where the squared [21]. We postulate the absence of coupling between two states which have, one, only non-strange quarks and, the other, only strange quarks. The φ and f′ couplings to non-strange mesons or baryons are indeed much weaker than the ω and f couplings.Google Scholar
  47. 32.
    G. Veneziano, Nuovo Cimento 57A, 190 (1968).ADSGoogle Scholar
  48. 33.
    S. Mandelstam, Phys. Rev. Letters 21, 1724 (1968).ADSCrossRefGoogle Scholar
  49. 34.
    M. Gell-Mann, Geneva Conference (1962).Google Scholar
  50. 35.
    K. Igi, Phys. Letters 28B, 330 (1968).ADSGoogle Scholar
  51. 36.
    E. Malamund and P. E. Schlein, Phys. Rev. Letters 18, 1056 (1967).ADSCrossRefGoogle Scholar
  52. C. Lovelace, R. M. Heinz and A. Donnachie, Phys. Letters 22B, 322 (1966).ADSGoogle Scholar
  53. 37.
    C. Lovelace, Phys. Letters 28B, 265 (1968).ADSGoogle Scholar
  54. 38.
    S. L. Adler, Phys. Rev. 137, B1022 (1965).ADSCrossRefGoogle Scholar
  55. 39.
    S. Weinberg, Phys. Rev. Letters 17, 616 (1966).ADSCrossRefGoogle Scholar
  56. 40.
    K. Kawarabayashi, S. Kitakado and H. Yabuki, Phys. Letters 28B, 432 (1969).ADSGoogle Scholar
  57. C. Lovelace, unpublished.Google Scholar
  58. 41.
    G. P. Canning, Oxford preprint (1968). J. Rosner, Tel Aviv preprint (1968); H. Olson, London preprint (1968); J. Baacke et al., CERN preprint TH.983 (1968).Google Scholar
  59. 42.
    F. Wagner, CERN preprint TH.978 (1969).Google Scholar
  60. 43.
    A. Martin, Nuovo Cimento 47A, 265 (1967).ADSGoogle Scholar
  61. 44.
    A. Montanet et al., CERN preprint (1969).Google Scholar
  62. 45.
    M. Ademollo, G. Veneziano, and S. Weinberg, Phys. Rev. Letters 22, 83 (1969).ADSCrossRefGoogle Scholar
  63. 46.
    The identity of the slopes is proved in Ref. [45]. Relation (30) can be also written for an external particle with the same spin SA on the assumed parallel daughter. This implies that the ratio of the two slopes is an integer. Assuming that the Adler condition is satisfied the same way (through vanishing of each term) when A and X replace one another, the inverse ratio has also to be an integer and as a result.Google Scholar
  64. 47.
    K. Igi and R. Storrow, to be published in Nuovo Cimento; C. Lovelace, to be published.Google Scholar
  65. 48.
    The analysis sketched here was done in collaboration with J. Baacke, S. Pokorsky and C. Schmid.Google Scholar
  66. 49.
    We disregard here the possibility that an isospin zero trajectory, at the pion daughter level, might “conspire” with the π A1, trajectory in order to eliminate exotic resonances from the crossed channel. We choose to relate together only leading trajectories in either channels.Google Scholar
  67. 50.
    P. G. O. Freund and E. Schonberg, Phys. Letters 28B, 600 (1969).ADSGoogle Scholar
  68. 51.
    V. de Alfaro, S. Fubini, G. Furlan and C. Rossetti, Phys. Letters 21, 576 (1966).ADSCrossRefGoogle Scholar
  69. F. Gilman and H. Harari, Phys. Rev. 165, 1803 (1968).ADSCrossRefGoogle Scholar
  70. 52.
    K. Bardakci and H. Ruegg, Phys. Letters 28B, 342 (1968).ADSGoogle Scholar
  71. 53.
    Chan Hong-Mo, Phys. Letters 28B, 425 (1969).Google Scholar
  72. Chan Hong-Mo and Tsou Sheung Tsun, Phys. Letters 28B, 485 (1969).Google Scholar
  73. J. F. L. Hopkinson and E. Plahte, Phys. Letters 28B, 489 (1969).ADSGoogle Scholar
  74. M. A. Virasoro, Phys. Rev. Letters 22, 37 (1969).ADSCrossRefGoogle Scholar
  75. J. G. Goebel and B. Sakita, Phys. Rev. Letters 22, 257 (1969).ADSCrossRefGoogle Scholar

Copyright information

© Springer-Verlag / Wien 1969

Authors and Affiliations

  • M. Jacob
    • 1
  1. 1.CERNGenevaSwitzerland

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