Il Nuovo Cimento A (1971-1996)

, Volume 57, Issue 2, pp 253–300 | Cite as

On the group-theoretical approach to the conspiracy problem for arbitrary masses.—I

  • G. Cosenza
  • A. Sciarrino
  • M. Toller


The scattering amplitude for an arbitrary two-particle process is written as a function\(T_{m_1 m_2 m_3 m_4 }^W (h)\) (h), whereW is the square of the four-momentum transfer andh is an element of a subgroupH W c of the complex Lorentz group. If suitable conventions are used, this function is shown to be free of kinematic singularities and free of kinematic constraints, apart from two covariance conditions of group-theoretical nature. The usefulness of this function in the treatment of the kinematic conditions in a Regge-pole model is shown and discussed. The matrix elements of the representations of the homogeneous Lorentz group are used in order to obtain explicit expressions for this function, which satisfy, above a certain energy and for small momentum transfer, all the requirements of analyticity and covariance. These expressions are equivalent to the contribution of a family of parallel conspiring Regge trajectories with well-defined quantum numbers and factorized residues. In this first part, only unequal-mass scattering is treated in detail. A detailed treatment of the equal-mass case and of the general factorization conditions will be given in the second part of this paper.

О теорико-групповом подходе к проблеме конспиративности для произвольных масс


Амплитуда рассеяния произвольного двухчастичного процесса записывается, как функция\(T_{m_1 m_2 m_3 m_4 }^W (h)\), гдеW представляет квадрат четырехмерного передаваемого импульса иh есть элемент подгруппыH W c для комплексной группы Лорентца. Если используются удобные обозначения, то показывается, что эта функция свободна от кинематических сингулярностей и срободна от кинематических ограничений, за исключением двух ковариантных условий, которые имеют теорикогрупповую природу. Поксзывается и обсуждается полезность этой функции в интерпретации кинематических условий в модели полюсов Редже. Матричные элементы для представлений однородной группы Лорентца используются, чтобы получить точные выражения для этой функции, которые удовлетворяют, при определенной энергии и для малых передаваемых импульсов, всем требованиям аналитичности и ковариантности. Эти выражения экввалентны вкладу семейства параллельных конспиративных траектории Редже с точно-определенными квантовыми числами и факторизуемыми вычетами. В этой первой части, подробно рассматривается толюко рассеяние для неравных масс. Подробное рассмотрение случая равных масс и общие условия факторизации будут приведены во второй части этой раборы.


L'ampiezza di diffusione per un arbitrario processo a due particelle viene scritta come una funzione\(T_{m_1 m_2 m_3 m_4 }^W (h)\) doveW è il quadrato del quadrimomento trasferito edh è un elemento del sottogruppoH W c del gruppo complesso di Lorentz. Si dimostra come, se si introducono opportune convenzioni, tale funzione risulti priva di singolarità e vincoli cinematici salvo che per due condizioni di covarianza di natura gruppale. Si discute poi l'utilità di questa funzione nello studio delle condizioni cinematiche in un modello a poli di Regge. Per ottenere espressioni esplicite di questa funzione che soddisfino, per piccolo momento trasferito ed al di sopra di una certa energia, tutti i requisiti di analiticità e covarianza, si usano gli elementi di matrice delle rappresentazioni del gruppo omogeneo di Lorentz. Queste espressioni sono equivalenti al contributo di una famiglia di traiettorie di Regge parallele e cospiranti con ben definiti numeri quantici e residui fattorizzati. In questa prima parte, si trattano in dettaglio solo processi con masse diseguali. Nella seconda parte di questo lavoro si tratteranno in dettaglio i casi in cui alcune masse sono eguali e le condizioni di fattorizzazione in generale.


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  1. (1).
    Y. Hara:Phys. Rev.,136, B 507 (1964).CrossRefADSMathSciNetGoogle Scholar
  2. (2).
    L. L. Wang:Phys. Rev.,142, 1187 (1966).CrossRefADSGoogle Scholar
  3. (3).
    H. P. Stapp:Phys. Rev.,160, 1251 (1967).CrossRefADSGoogle Scholar
  4. (4).
    H. Högaasen andPh. Salin:Nucl. Phys., B2, 657 (1967).CrossRefADSGoogle Scholar
  5. (5).
    G. Cohen-Tannoudji, A. Morel andH. Navelet:Ann. of Phys.,46, 239 (1968).CrossRefADSGoogle Scholar
  6. (6).
    B. Diu andM. Le Bellac:Nuovo Cimento,53 A, 158 (1968).CrossRefADSGoogle Scholar
  7. (7).
    S. Frautschi andL. Jones:Phys. Rev.,163, 1820 (1967);164, 1918 (1967);167, 1335 (1968).CrossRefADSGoogle Scholar
  8. (8).
    D. Z. Freedman andJ. M. Wang:Phys. Rev. Lett.,17, 569 (1966);Phys. Rev.,153, 1596 (1967).CrossRefADSGoogle Scholar
  9. (9).
    D. Z. Freedman, C. E. Jones andJ. M. Wang:Phys. Rev.,155, 1645 (1967).CrossRefADSGoogle Scholar
  10. (10).
    M. L. Goldberger andC. E. Jones:Phys. Rev. Lett.,17, 105 (1966);Phys. Rev.,150, 1269 (1966).CrossRefADSGoogle Scholar
  11. (11).
    L. Durand III:Phys. Rev. Lett.,18, 58 (1967);Phys. Rev.,154, 1537 (1967).CrossRefADSGoogle Scholar
  12. (12).
    D. V. Volkov andV. N. Gribov:Žurn. Ėksp. Teor. Fiz.,44, 1068 (1963). English translation:Sov. Phys. JETP,17, 720 (1963).Google Scholar
  13. (13).
    V. N. Gribov:Žurn. Ėksp. Teor. Fiz.,43, 1529 (1962), English translation:Sov. Phys. JETP,16, 1080 (1963).Google Scholar
  14. (14).
    V. N. Gribov, L. Okun' andI. Pomeranchuk:Žurn. Ėksp. Teor. Fiz.,45, 1114 (1963), English translation:Sov. Phys. JETP,18, 769 (1964).Google Scholar
  15. (15).
    E. Leader:Phys. Rev.,166, 1599 (1968).CrossRefADSGoogle Scholar
  16. (16).
    G. C. Fox andE. Leader:Phys. Rev. Lett. 18, 628 (1967).CrossRefADSGoogle Scholar
  17. (17).
    M. Gell-Mann andE. Leader:XIII International Conference on High-Energy Physics, Berkeley, 1967.Google Scholar
  18. (18).
    J. C. Taylor:Regge poles in invariant amplitudes and families of trajectories, Oxford preprint (1967);Nucl. Phys., B3, 504 (1967).Google Scholar
  19. (19).
    For a complete list of references about recent developments of Regge-pole theory and phenomenology, see:L. Bertocchi:Proceedings of the Heidelberg International Conference on Elementary Particles,H. Filthuth Editor (Amsterdam, 1968), p. 197.Google Scholar
  20. (20).
    Many Authors (15). do not use the term «conspiracy» in this case. We prefer to use a more general definition of «conspiracy».CrossRefADSGoogle Scholar
  21. (21).
    M. Toller:Nuovo Cimento,53 A, 671 (1968). References to previous work can be found in this paper.CrossRefADSGoogle Scholar
  22. (22).
    A. Sciarrino andM. Toller:Journ. Math. Phys.,8, 675 (1967).CrossRefMathSciNetGoogle Scholar
  23. (23).
    G. Domokos:Phys. Rev.,159, 1387 (1967);Phys. Lett.,24 B, 293 (1967).CrossRefADSGoogle Scholar
  24. (24).
    D. Z. Freedman andJ. M. Wang:Phys. Rev. Lett.,18, 863 (1967);Phys. Rev.,160, 1560 (1967).CrossRefADSGoogle Scholar
  25. (25).
    H. Joos:Lectures in Theoretical Physics, vol.7 A, edited byW. E. Brittin andA. O. Barut (Boulder, 1965), p. 132.Google Scholar
  26. (26).
    M. Toller:Nuovo Cimento,37, 631 (1965);54 A, 295 (1968). In the last paper a complete list of references can be found.CrossRefMathSciNetGoogle Scholar
  27. (27).
    M. Sheftel, J. Smorodinskij andP. W. Winternitz:Phys. Lett.,26 B, 241 (1968). In this paper a complete list of references and discussion of previous papers is given.CrossRefADSGoogle Scholar
  28. (28).
    E. P. Wigner:Ann. Math.,40, 149 (1939).CrossRefADSMathSciNetGoogle Scholar
  29. (29).
    H. Joos:Fortschr. d. Phys.,10, 65 (1962).CrossRefADSGoogle Scholar
  30. (30).
    P. Moussa andR. Stora: inLectures in Theoretical Physics, vol.7 A, edited byW. E. Brittin andA. O. Barut (Boulder, 1965), p. 37.Google Scholar
  31. (31).
    For instance, the solutions given by (12), for nucleon-nucleon scattering are not single Lorentz poles. However it is possible that, considering reactions with arbitrarily high spins and taking into account the factorization condition, solutions which are not Lorentz poles can be rejected.Google Scholar
  32. (32).
    G. C. Wick:Phys. Rev.,96, 1124 (1954).CrossRefADSMathSciNetGoogle Scholar
  33. (33).
    D. Amati, S. Fubini andA. Stanghellini:Nuovo Cimento,26, 896 (1962).CrossRefGoogle Scholar
  34. (34).
    C. Ceolin, F. Duimio, R. Stroffolini andS. Fubini:Nuovo Cimento,26, 247 (1962).CrossRefGoogle Scholar
  35. (35).
    D. Amati, A. Stanghellini andK. Wilson:Nuovo Cimento,28, 539 (1963).CrossRefMathSciNetGoogle Scholar
  36. (36).
    G. Domokos andP. Surányi:Nucl. Phys.,54, 529 (1964).CrossRefGoogle Scholar
  37. (37).
    J. D. Bjorken:Journ. Math. Phys.,5 192 (1964).CrossRefADSMathSciNetGoogle Scholar
  38. (38).
    It is possible that this does not happen for solutions of the «simplified problem» which are not Lorentz poles, if such solutions exist (see note (31))..Google Scholar
  39. (39).
    R. F. Sawyer:Phys. Rev. Lett.,18, 1212 (1967);19, 137 (1967);Phys. Rev.,167, 1372 (1968).CrossRefADSMathSciNetGoogle Scholar
  40. (40).
    P. K. Mitter:Phys. Rev.,162, 1624 (1967);O 4 symmetry and the forward production of vector mesons at high energies. University of California, Santa Barbara, preprint (1967).CrossRefADSGoogle Scholar
  41. (41).
    R. J. Oakes:Phys. Lett.,24 B, 154 (1967).CrossRefADSGoogle Scholar
  42. (42).
    R. Delbourgo, A. Salam andJ. Strathdee:Phys. Lett.,25 B, 230 (1967);Phys. Rev.,164, 1981 (1967).CrossRefADSGoogle Scholar
  43. (43).
    G. Domokos andG. L. Tindle:Phys. Rev.,165, 1906 (1968);Comm. Math. Phys.,7, 160 (1967).CrossRefADSGoogle Scholar
  44. (44).
    E. Inonu andE. P. Wigner:Proc. Nat. Acad. Sci.,39, 510 (1953).CrossRefADSMathSciNetGoogle Scholar
  45. (45).
    E. J. Saletan:Journ. Math. Phys.,2, 1 (1961).CrossRefADSMathSciNetGoogle Scholar
  46. (46).
    R. Hermann:Lie Groups for Physicists (New York, 1966).Google Scholar
  47. (47).
    R. Hermann:Fourier analysis on groups and partial wave analysis, I, University of California, Berkeley, preprint (1967);Comm. Math. Phys.,6, 205 (1967), where reference to previous work is given.Google Scholar
  48. (48).
    Of course, this expansion is meaningful only in a subset of the complex Lorentz group, which contains the real Lorentz group and the physically relevant region. Another possibility could be to expand the function in terms of representations of the complex Lorentz group, generalizing a formalism developed by (49)..CrossRefADSMathSciNetGoogle Scholar
  49. (49).
    E. H. Roffman:Phys. Rev. Lett.,16, 210 (1966);Comm. Math. Phys.,4, 237 (1967);Phys. Rev.,167, 1424 (1968).CrossRefADSMathSciNetGoogle Scholar
  50. (50).
    R. F. Streater andA. S. Wightman:PCT, Spin and Statistics and All That (New York, 1964).Google Scholar
  51. (51).
    H. P. Stapp:Phys. Rev.,125, 2139 (1962); and inHigh-Energy Physics and Elementary particles (Trieste Seminars, 1965), p. 3 (International Atomic Energy Agency, Vienna, 1965).CrossRefADSMathSciNetGoogle Scholar
  52. (52).
    D. Hall andA. S. Wightman:Kgl. Danske Videnskab. Selskab, Mat. Fys. Medd.,31, No. 5 (1957).Google Scholar
  53. (53).
    A. O. Barut, I. Muzinich andD. N. Williams:Phys. Rev.,130, 442 (1963).CrossRefADSMathSciNetGoogle Scholar
  54. (54).
    D. N. Williams: Berkeley report UCRL 11113 (1963).Google Scholar
  55. (55).
    K. Hepp:Helv. Phys. Acta,36, 355 (1963).MathSciNetGoogle Scholar
  56. (56).
    M. Jacob andG. C. Wick:Ann. of Phys.,7, 404 (1959).CrossRefADSMathSciNetGoogle Scholar
  57. (57).
    M. Gell-Mann, M. L. Goldberger, F. E. Low, E. Marx andF. Zachariasen:Phys. Rev.,133, B 145 (1964).CrossRefADSMathSciNetGoogle Scholar
  58. (58).
    F. Calogero, J. M. Charap andE. J. Squires:Ann. Phys.,25, 325 (1963).CrossRefADSMathSciNetGoogle Scholar
  59. (59).
    W. Drechsler:Nuovo Cimento,53 A, 115 (1968).CrossRefADSGoogle Scholar
  60. (60).
    H. Behnke andP. Thullen:Theorie der Funktionen mehrerer komplexer Veränderlicher (Berlin, 1934), p. 50.Google Scholar
  61. (61).
    In order to use Theorem 3, we have to map analytically a neighbourhood of the point we are considering onto a domain ofC 20 and to show that the set where the conditions (3.35) and (3.39) do not hold is mapped into a 17-dimensional complex manifold. This is possible if the point we are considering does not belong to the set defined byP 1/M 1=±P 2/M 2=±P 3/M 3=±P 4/M 4,P i=L(a i)P o(i). Using a more refined argument, it is possible to continue analytically theM-function also in this singular set. We are grateful to Dr.H. Epstein for his precious advice, which has permitted the completion of this proof.Google Scholar
  62. (62).
    N. N. Khuri:Phys. Rev. Lett.,10, 420 (1963);Phys. Rev.,132, 914 (1963).CrossRefADSMathSciNetGoogle Scholar
  63. (63).
    R. E. Cutkoski andB. B. Deo:Phys. Rev. Lett.,19, 1256 (1967).CrossRefADSGoogle Scholar
  64. (64).
    L. Sugar andJ. D. Sullivan:Phys. Rev.,166, 1515 (1968).CrossRefADSGoogle Scholar
  65. (65).
    M. E. Rose:Elementary Theory of Angular Momentum (New York, 1957).Google Scholar
  66. (66).
    For a review article on induced representations which contains references to the original works, see the AppendixGroup Representations in Hilbert Spaces, byG. Mackey inI. E. Segal:Mathematical Problems in Relativistic Quantum Mechanics (Providence, R.I., 1963).Google Scholar
  67. (67).
    M. A. Naimark:Linear Representations of Lorentz Group (London, 1964).Google Scholar
  68. (68).
    A. Erdelyi, W. Magnus, F. Oberhettinger andF. G. Tricomi:Higher Transcendental Functions, vol.1 (New York, 1953).Google Scholar
  69. (69).
    D. A. Akyeampong, J. F. Boyce andM. A. Rashid:Nuovo Cimento,53 A, 737 (1968).CrossRefADSGoogle Scholar

Copyright information

© Società Italiana di Fisica 1968

Authors and Affiliations

  • G. Cosenza
    • 1
    • 2
  • A. Sciarrino
    • 2
    • 3
  • M. Toller
    • 4
  1. 1.Istituto di Fisica dell' UniversitàNapoli
  2. 2.Istituto di Fisica dell' UniversitàRoma
  3. 3.Scuola di Perfezionamento in FisicaRoma
  4. 4.CERNGeneva

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