Skip to main content

A PCAC puzzle: π0→γγ in the σ-model

Затруднения РСАС: π0→γγ в σ модели


The effective coupling constant for π0→γγ should vanish for zero pion mass in theories with PCAC and gauge invariance. It does not so vanish in an explicit perturbation calculation in the σ-model. The resolution of the puzzle is effected by a modification of Pauli-Villars-Gupta regularization which respects both PCAC and gauge invariance.


La costante di accoppiamento effettiva per π0→γγ si dovrebbe annullare per massa del pione nulla nelle teorie con PCAC e invarianza di gauge. Non si dovrebbe verificare la stessa cosa in un esplicito calcolo perturbativo nel modello σ. Si risolve il rompicapo modificando la regolarizzazione di Pauli. Villars e Gupta, che rispetta sia la PCAC che l’invarianza di gauge.


Зффективная константа связи для π0→γγ должна обращаться в нуль при нулевой массе пиона в теориях с РСАС и калибровочной инвариантностыо. Однако, она не обращается в нуль при точном пертурбационном вычислении в σ модели. Сазрешение зтого затруднения осуществляется лутем видоизменения регуляризации Паули-Вилларса-Гупта, которая учитывает и РСАС и калибровочную инвариантность.

This is a preview of subscription content, access via your institution.


  1. J. Steinberger:Phys. Rev.,76, 1180 (1949).

    Article  ADS  Google Scholar 

  2. D. G. Sutherland:Nucl. Phys., B2, 433 (1967).

    Article  ADS  Google Scholar 

  3. T. Goto andI. Imamura:Progr. Theor. Phys.,14, 396 (1955);J. Schwinger:Phys. Rev. Lett.,3, 296 (1959).

    Article  ADS  Google Scholar 

  4. K. Johnson andF. E. Low:Progr. Theor. Phys. (Kyoto) Suppl., No.37-88, 74 (1966). In connection with this paper see alsoB. Hamprecht:Nuovo Cimento,50A, 449 (1967);J. C. Polkinghorne:Nuovo Cimento 52A, 351 (1967). These authors observe that the commutators as calculated byJohnson andLow depend on the order in which certain limits are performed. It was explained in (5) that different orders are appropriate for different applications.

  5. J. S. Bell:Nuovo Cimento,47A, 616 (1967). The author of this paper takes the opportunity to refute a criticism of it (6). It was observed in (5) that even when certain sum rules involve «effective» values for certain commutators, the «canonical» values remain appropriate for certain propagator identities or zero-energy theorems. This depends on defining the propagators as limits of those in a suitable cut-off theory. It was also noted that other objects, referred to as «mutilated» propagators, differing from the originals by contact terms, satisfy identities involving the «effective» noncanonical commutators. In (6) a third way of defining propagators is considered; it is in certain circumstances, and in particular for the disputed model, equivalent to «mutilation». It is then contended that only the latter definition has physical interest. The reasons appear to be a), a dislike for cut-off, and b), a preference for quantities «carefully defined from a distribution-theoretic point of view». With regard to a), it would indeed be better to avoid cut-offs if possible; but the reference cited in this connection (R. Haag andG. Luzatto:Nuovo Cimento,13, 415 (1959)) is not relevant. For the final formulation there involves only the renormalized coupling constant, and reduces trivially to free fields when this quantity is zero—as it is in the limit studied in (5,6). With regard to b), it is rather well known that arbitrarily definedT products may differ from physical amplitudes by contact terms; see for example the discussion ofBjorken referred to in (5). The positive reasons for taking, some interest in the limiting propagators, without mutilation, were stated in (5). They are defined (and how else could one find quantities of «physical» interest in such a model?) in analogy with quantities of interest in more serious theories. Explicit reference was made to Pauli-Villars regularization of the vacuum polarization tensor of electrodynamics.

    Article  ADS  Google Scholar 

  6. R. A. Brandt andC. A., Orzalesi:Phys. Rev.,162, 1747 (1967).

    Article  ADS  Google Scholar 

  7. A very useful and well-documented account of the application of regularization to the related problems of photon-mass and the Goto-Imamura-Schwinger terms has been given byG. Källén:Lectures Given at the Winter Schools in Karpacz, and Schladming, Feb. and Mar. 1968.

  8. M. Gell-Mann andM. Lévy:Nuovo Cimento,16, 705 (1960).

    Article  Google Scholar 

  9. S. N. Gupta:Proc. Phys. Soc., A66, 129 (1953).

    Article  ADS  Google Scholar 

  10. One of us (J.S.B.) acknowledges a very useful conversation withN. Kroll on the Gupta method.

Download references

Author information

Authors and Affiliations


Additional information

Traduzione a cura della Redazione.

Переведено редакцией.

Rights and permissions

Reprints and Permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bell, J.S., Jackiw, R. A PCAC puzzle: π0→γγ in the σ-model. Nuovo Cimento A (1965-1970) 60, 47–61 (1969).

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: