Summary
A field-theoretical proof is given of a relativistic generalization of Levinson’s theorem. Though the final form of this relation is in disagreement with an earlier attempt in this direction by Charap, a complete correspondence is shown to exist between the result obtained here and that derived by Warnock on the basis of dispersion theory. It is demonstrated that the discrepancy between Charap’s approach and that presented here originates both in the technique used for the treatment of bound states as well as in the method of counting constraint equations and CDD poles. As a test of this work it is shown that there is complete consistency in the case of the Zachariasen model, independent of whether the bound state is elementary or composite. As an auxiliary result a useful representation for the reciprocal of the usual denomination function of dispersion theory is derived which includes the effect of inelastic channels.
Riassunto
Si espone la prova in teoria dei campi di una generalizzazione relativistica del teorema di Levinson. Sebbene la forma finale di questa relazione sia in disaccordo con un precedente tentativo di Charap in questa direzione, si dimostra che esiste una completa corrispondenza fra il risultato ottenuto qui e quello dedotto da Warnock sulla base della teoria della dispersione. Si dimostra che la discrepanza fra l’approccio di Charap e quello presentato qui deriva sia dalla tecnica usata per il trattamento degli stati legati che dal metodo di contare le equazioni di costrizione e i poli di CDD. Come prova di questa trattazione si mostra che c’è una completa coerenza nel caso del modello di Zachariasen, indipendentemente dal fatto che lo stato legato sia elementare o composto. Come risultato ausiliario si deduce una utile rappresentazione della reciproca dell’usuale funzione di denominazione della teoria della dispersione che include l’effetto dei canali inelastici.
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References
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RecallingD(s)=[Δ(s)Γ(s)]−1 one sees that the asymptotic behaviour (13) implies the existence of a CDD pole inD(s) at infinity, thus providing the motivation for (14) in whichn c is represented as the totality of such poles in the extended plane. It should be remarked that while this notation is consistent with that of Warnock (5)Charap has failed to include the pole at infinity in his definition ofn c . Since, however, the presence of logarithmic factors could obscure the identification of this singularity ofD(s) at infinity as a pole, it may be convenient to avoid such complications by consideringD(s)/(s−a 2),i.e. by the elimination of the asymptotic linear divergence in favor of a CDD pole in the finite plane.
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Research supported in part by the U. S. Atomic Energy Commission.
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Hagen, C.R. A field-theoretical formulation of Levinson’s theorem. Nuovo Cimento A (1965-1970) 43, 597–614 (1966). https://doi.org/10.1007/BF02756683
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DOI: https://doi.org/10.1007/BF02756683