Il Nuovo Cimento A (1965-1970)

, Volume 8, Issue 3, pp 525–535 | Cite as

Quark spin in a dual-resonance model

  • P. G. O. Freund


The foundations are laid for a dual-resonance model with a spectrum characteristic ofU6×U6×O3 symmetry. The model provides an automatic mechanism for the breaking of the collinearU6×O2 symmetry. The states on the leading Regge trajectory with the exception of the lowest (« ground ») state are all parity doubled. It is argued that there may exist « mesonic » strings with a quark at one end and anSU3-singlet spin-zero boson at the other end. These complex hadrons would have all the quantum numbers (half-integer spin, nonvanishing triality, etc.) of quarks, while not being really quarks (e.g., a baryon would not be built of three of them).

Спин кварков в дуальной реэонансной модели


Формулируются обоснования для дуальной реэонансной модели со спектром, характерным дляU6 ×U6 ×О3 симметрии. Эта модель обеспечивает автоматический механиэм для нарущения коллинеарнойU6 ×O2 симметрии. Состояния на главной траектории Редже, эа исключением нижнего (основного) состояния, характериэуются удвоением четности. Докаэывается, что могут сушествовать « меэонные « цепочки с кварком на одном конце иSU3-ащглетным боэоном с нулевым спином на другом конце. Эти комплексные адроны имеют все квантовые числа (полуцелый спин, ненулевая триальность и т.д.) кварков, в то же время они не являются, в действительности, кварками (т.е. барион не может быть построен иэ трех таких адронов).


Si pongono le fondamenta di un modello di risonanza duale con uno spettro caratteristico della simmetriaU6×U6×O3. Il modello fornisce un meccanismo automatico per la rottura della simmetriaU6×O2 collineare. Gli stati sulle traiettorie di Regge principale, eccetto lo stato inferiore (« fondamentale ») sono tutti raddoppiati in parità. Si argomenta che possono esistere catene « mesoniche » con un quark ad una estremità e un bosone di spin 0 del singoletto diSU3 all’altra estremità. Questi adroni complessi avrebbero tutti i numeri quantici (spin semiintero, trialità non tendente a zero, ecc.) dei quark, pur non essendo in realtà quark (per esempio, un barione non sarebbe costruito con tre di essi).


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. (1).
    M. Gell-Mann:Phys. Lett.,8, 214 (1964);G. Zweig: CERN report (1964), unpublished.CrossRefADSGoogle Scholar
  2. (2).
    See,e.g., the review byP. G. O. Freund:Proceedings of the Boulder Conference on High-Energy Physics, edited byK. T. Mahanthappa et al. (Boulder, 1970), p. 563.Google Scholar
  3. (3).
    G. Veneziano:Nuovo Cimento,57 A, 190 (1968).CrossRefADSGoogle Scholar
  4. (4).
    S. Fubini andG. Veneziano:Nuovo Cimento,64 A, 811 (1969);K. Bardakci andS. Mandelstam:Phys. Rev.,184, 1640 (1969).CrossRefADSGoogle Scholar
  5. (5).
    Y. Nambu:Proceedings of the International Conference on Symmetries and Quark Models (Detroit, Mich., 1969).L. Susskind:Nuovo Cimento,69 A, 457 (1970).Google Scholar
  6. (6).
    K. Bardakci andM. B. Halpern:Phys. Rev. Lett.,24, 186 (1970).CrossRefGoogle Scholar
  7. (7).
    R. Carlitz, S. Ellis, P. G. O. Freund andS. Matsuda: Caltech preprint CALT 68-260, June 1970, unpublished.Google Scholar
  8. (8).
    P. Ramond:Phys. Rev. D,3, 2415 (1971).MathSciNetCrossRefADSGoogle Scholar
  9. (9).
    A. Neveu andJ. Schwarz: Princeton University preprint, March 1971.Google Scholar
  10. (10).
    A. Neveu andJ. Schwarz: Princeton University preprint, PURC-4159-27.Google Scholar
  11. (11).
    A. Neveu, J. Schwarz andC. B. Thorn: Princeton University preprint.Google Scholar
  12. (12).
    C. B. Thorn: Berkeley preprint.Google Scholar
  13. (13).
    M. B. Halpern andC. B. Thorn: Berkeley preprint.Google Scholar
  14. (*).
    In detail, the scheme of ref. (6) is not factorizable; the schemes of ref. (8–13) lead to parity doubling along the leading Regge trajectories with the exception of the lowest (ground) states on this trajectory. They are also incompatible with exactSU 3 symmetry as they require the unphysical relation αρ(0)−αω(0)=1/2. The model of ref. (7) enforces exactU 6W×O 2Lz symmetry at all vertices and leads to no parity doubling. The price paid for this is the appearance of a continuum (i.e. nonresonant) part of the spectrum. This continuum is dual to (i.e. saturates) the singular part of the Regge terms (U 6×O 2 requires Regge residues that become singular att=0) and the corresponding Carlitz-Kislinger cuts that restore the meromorphy ins andt of the complete amplitude.CrossRefGoogle Scholar
  15. (14).
    P. G. O. Freund, A. Maheshwari andE. Schonberg:Phys. Rev.,159, 1232 (1967);H. J. Lipkin:Phys. Rev.,159, 1303 (1967).CrossRefADSGoogle Scholar
  16. (15).
    S. Fubini, D. Gordon andG. Veneziano:Phys. Lett.,29 B, 679 (1969).CrossRefADSGoogle Scholar
  17. (16).
    M. Virasoro:Phys. Rev. D,1, 2933 (1970).CrossRefADSGoogle Scholar
  18. (17).
    A. Salam, R. Delbourgo andJ. Strathdee:Proc. Roy. Soc., A284, 146 (1965);B. Sakita andK. C. Wali:Phys. Rev.,139, B 1355 (1965).MathSciNetCrossRefADSGoogle Scholar
  19. (18).
    P. G. O. Freund:Phys. Lett.,15, 352 (1965);M. Y. Han andY. Nambu:Phys. Rev.,139, B 1006 (1965).CrossRefADSGoogle Scholar
  20. (19).
    P. G. O. Freund andJ. L. Rosner: unpublished.Google Scholar
  21. (20).
    Y. Dothan, M. Gell-Mann andY. Ne’eman:Phys. Lett.,17, 488 (1965).MathSciNetCrossRefGoogle Scholar
  22. (21).
    See,e.g.,C. S. Guralnik andT. W. B. Kibble:Phys. Rev.,139, B 712 (1965).MathSciNetCrossRefADSGoogle Scholar
  23. (22).
    Y. Aharonov, A. Casher andL. Susskind: Tel-Aviv University preprint TAUP-212-71.Google Scholar
  24. (23).
    H. Harari:Phys. Rev. Lett.,22, 562 (1969);J. L. Rosner:Phys. Rev. Lett.,22, 689 (1969);T. Matsuoka, K. Ninomiya andS. Sawada:Progr. Theor. Phys.,42, 56 (1969).CrossRefADSGoogle Scholar
  25. (24).
    P. G. O. Freund:Nucl. Phys.,29 B, 317 (1971).CrossRefADSGoogle Scholar
  26. (25).
    P. G. O. Freund, R. E. Waltz andJ. L. Rosner:Nucl. Phys.,13 B, 237 (1969).CrossRefADSGoogle Scholar
  27. (*).
    After the completion of this work, I learned from Prof.S. Mandelstam about new work on spin in dual-resonance models byK. Bardakci andM. B. Halpern (26). They use a different formalism from that of this paper. This permits them to obtain simple expressions for the vertices, but no wave equations. In our approach the situation is just the opposite.Google Scholar
  28. (26).
    K. Bardakci: Berkeley preprint;M. B. Halpern: Berkeley preprint.Google Scholar

Copyright information

© Società Italiana di Fisica 1972

Authors and Affiliations

  • P. G. O. Freund
    • 1
    • 2
  1. 1.Department of PhysicsImperial CollegeLondon
  2. 2.The Enrico Fermi Institute and the Department of PhysicsThe University of ChicagoChicago

Personalised recommendations