Il Nuovo Cimento A (1965-1970)

, Volume 11, Issue 2, pp 397–435 | Cite as

A renormalizable gauge model of lepton interactions

  • Abdus Salam
  • J. Strathdee


It is known that the spontaneous violation of a gauge symmetry of the second kind results in the appearance not of Goldstone bosons but, rather, of massive gauge particles. The path-integral quantization of these theories is discussed here in general terms. The primary consideration is that quantities of physical significance, such as matrix elements of the scattering operator or the energy-momentum tensor, should be independent of the gauge in which the quantization rules are formulated. In particular, if it is possible to find one gauge in which the theory is unitary and another in which it is renormalizable, then the gauge-independent quantities must enjoy both these qualities. These ideas are applied to simple models with massive Yang-Mills fields and to a model which unifies the weak and electromagnetic interactions of electron-type leptons. Both these models appear to be unitary and renormalizable. The lepton theory is a relatively economical one. It involves five independent parameters: the electron charge and mass, the mass of the charged intermediate vector boson and the masses of a neutral scalar and a neutral vector boson.

Перенормируемая калибровочная модель лептонных вэаимодействий


Иэвестно, что спонтанное нарущение калибровочной симметрии второго рода приводит к появлению не боэонов Голдстоуна, а скорее массивных калибровочных частиц. В зтой работе в обших выражениях обсуждается квантование с помошью контурных интегралов таких теорий. Исходное соображение состоит в том, что величины, имеюшие фиэический смысл, такие как матричные злементы оператора рассеяния или тенэор знергии-импульса, должны быть не эависимыми от калибровки, в которой формулируются правила квантования. В частности, если воэможно найти одну калибровку, в которой теория является унитарной, и другую калибровку, в которой теория является перенормируемой, тогда не эависяшие от калибровки величины должны обладать обоими качествами. Эти идеи применяются к простой модели с массивными полями Янга-Миллса и к модели, которая общединяет слабые и злектромагнитные вэаимодействия лептонов злектронного типа. Обе зти модели окаэываются унитарными и перенормируемыми. Лептонная теория является относительной зкономной. Эта теория включает пять неэависимых параметров: эаряд и массу злектрона, массу эаряженного промежуточного векторного боэона и массы нейтрального скалярного и нейтрального векторного боэонов.


È noto che la violazione spontanea di una simmetria di gauge di seconda specie produce l’apparizione non di bosoni di Goldstone ma, piuttosto, di particelle di gauge con massa. Si discute qui in termini generali la quantizzazione dell’integrale di percorso di queste teorie. La considerazione primaria è che le grandezze di significato fisico, como gli elementi di matrice dell’operatore di scattering o il tensore energia-impulso, devono essere indipendenti dal gauge in cui si formulano le regole di quantizzazione. In particolare, se è possible trovare un gauge in cui la teoria è unitaria ed un altro in cui essa è rinormalizzabile, allora le grandezze indipendenti dal gauge devono godere di entrambe queste qualità. Si applicano queste idee ad un semplice modello con campi di Yang-Mills con massa e ad un modello che unifica le interazioni deboli ed elettromagnetiche dei leptoni del tipo dell’elettrone. Entrambi questi modelli risultano unitari e rinormalizzabili. La teoria del leptone è relativamente economica. Essa comporta cinque parametri indipendenti: la massa e la carica dell’elettrone, la massa del bosone vettoriale intermedio carico, e le masse di un bosone scalare neutro e di un bosone vettoriale neutro.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. (1).
    G. Jona-Lasinio andY. Nambu:Phys. Rev.,122, 345 (1961). The same mechanism was proposed for self-consistently generating the electron mass in aγ 5-invariant theory byA. Salam:On Fermi interactions, Imperial College print (1957).CrossRefADSGoogle Scholar
  2. (2).
    J. Goldstone:Nuovo Cimento,19, 154 (1961);J. Goldstone, A. Salam andS. Weinberg:Phys. Rev.,127, 965 (1962).MathSciNetCrossRefzbMATHGoogle Scholar
  3. (3).
    P. Higgs:Phys. Rev.,145, 1156 (1966).MathSciNetCrossRefADSGoogle Scholar
  4. (4).
    T. W. B. Kibble:Phys. Rev.,155, 1554 (1967).CrossRefADSGoogle Scholar
  5. (5).
    G. t’Hooft:Renormalizable Lagrangians for massive Yang-Mills fields, Utrecht preprint (1971).Google Scholar
  6. (6).
    The model we consider was proposed byS. Weinberg (Phys. Rev. Lett.,19, 1264 (1967)) and independently byA. Salam: inProceedings of the Eighth Nobel Symposium, edited byN. Svartholm (Stockholm, 1968), p. 367.CrossRefADSGoogle Scholar
  7. (7).
    The model of ref. (6) possesses two ingredients; one is a unified vector-axial-vectorgauge set of terms combining weak and electromagnetic interactions and the second is Higgs-Kibble’s set of scalar fields. So far as the gauge Lagrangian combining weak and electromagnetic interactions is concerned, the history of attempts to devise these Lagrangians is a long one. See for example:J. Schwinger:Ann. of Phys.,2, 407 (1957);S. L. Glashow:Nucl. Phys.,10, 103 (1959);22, 579 (1961);A. Salam andJ. C. Ward:Nuovo Cimento,11, 568 (1959);Phys. Lett.,13, 168 (1964). In the last reference the left- and right-handed leptons ψ, e and μ+ were assigned to a leptonicU 3×U 3, though only a particularly economicalU 2 subgroup of this larger group was gauged.Weinberg in ref. (6) worked with the sameU 2, though in a recent MIT preprint (to be published inPhys. Rev.) he has proposed to gauge the fullU 3×U 3. The same suggestion has been made byFreund (ref. (13)). In this paper we shall confine our attention to the leptonicU 2 set out in detail in Sect.4, which is the same as in Salam and Ward’s paper (1964) (this reference) and the paper ofWeinberg (1967) (ref. (6)).CrossRefADSGoogle Scholar
  8. (8).
    A detailed examination of the renormalization programme in the context of a spontaneously violated Abelian gauge symmetry (scalar electrodynamics) has recently been carried out byB. W. Lee:Renormalizable massive vector meson theory-perturbation theory of the Higgs phenomenon, Batavia preprint (1971). (This paper was brought to our attention byA. Maheshwari while the present work was being completed.)Google Scholar
  9. (9).
    L. D. Faddeev andV. N. Popov:Phys. Lett.,25 B, 29 (1967); Kiev preprint ITF-67-36 (1967) (in Russian);L. D. Faddeev:Theoret. and Math. Phys.,1, 1 (1969).CrossRefADSGoogle Scholar
  10. (*).
    While this paper was in preparation, we received a preprint fromD. Gross andR. Jackiw, who argue that a serious difficulty in implementing the renormalizability of the theory discussed in this paper arises from the Schwinger-Bell-Jackiw-Adler anomaly associated with AVV and AAA vertices in the theory. They have also suggested that such difficulties can be removed by enlarging the Lagrangian to include hadrons also, in such a manner that the corresponding hadronic and leptonic anomalies cancel each other so far as weak and electromagnetic interactions are concerned. It is not clear, then, what the effect of introducing strong interactions is likely to be. We are grateful toR. Jackiw andD. Gross for communicating their results to us before publication.Google Scholar
  11. (10).
    R. P. Feynman:Acta Phys. Polon.,24, 697 (1963); see alsoR. N. Mohapatra:Phys. Rev. D,4, 378, 1007 (1971); University of Maryland Tech. Rep. 72-047.MathSciNetzbMATHGoogle Scholar
  12. (11).
    E. S. Fradkin andI. V. Tyutin:Phys. Rev. D,2, 2841 (1970).MathSciNetCrossRefADSGoogle Scholar
  13. (12).
    A. Salam andJ. Strathdee:Phys. Rev. D,2, 2869 (1970).CrossRefADSGoogle Scholar
  14. (*).
    This theory was treated byt’Hooft (ref. (5)) but in a gauge different from the ones considered here.Google Scholar
  15. (13).
    P. G. O. Freund:Leptonic and hadronic symmetries, University of Chicago preprint, EFI 72-03;A. Salam andJ. C. Ward:Phys. Lett.,13, 168 (1964);S. Weinberg: M.I.T. preprint (1972).Google Scholar
  16. (14).
    T. D. Lee:Phys. Rev. Lett.,26, 801 (1971);J. Schechter andY. Ueda:Phys. Rev. D,2, 736 (1967).CrossRefADSGoogle Scholar
  17. (15).
    D. Bessis andJ. Zinn-Justin:One-loop renormalization of the nonlinear σ-model, Saclay preprint (1971).Google Scholar
  18. (16).
    D. G. Boulware:Ann. of Phys.,56, 140 (1970).MathSciNetCrossRefADSGoogle Scholar
  19. (17).
    See, for example,A. Salam:Computation of renormalization constants, inProceedings of Coral Gables Conference, 1971.Google Scholar
  20. (18).
    C. J. Isham, A. Salam andJ. Strathdee:Phys. Rev. D,3, 1805 (1971); ICTP, Trieste, preprint IC/71/14 (to appear inPhys. Rev.).MathSciNetCrossRefADSGoogle Scholar

Copyright information

© Società Italiana di Fisica 1972

Authors and Affiliations

  • Abdus Salam
    • 1
    • 2
  • J. Strathdee
    • 3
  1. 1.International Centre for Theoretical PhysicsTrieste
  2. 2.Imperial CollegeLondon
  3. 3.International Centre for Theoretical PhysicsTrieste

Personalised recommendations