Summary
The analytic properties of the partial wave scattering amplitude for potential scattering in the pair of variablesk (wave number) andλ=l+1/2 have been derived when both variables are complex. Several results on the location of the poles of theS-matrix follow from a procedure of analytic completion. The scattering process is then considered as described by the variablesλ andk, instead ofs andt, as in Mandelstam work. The set of properties ofS(λ,k)=exp [2iδ(λ,k)] here derived is exactly equivalent to the double dispersion formulas for energy and momentum transfer.
Riassunto
Vengono ricavate le proprietà analitiche dell’ampiezza di diffusione in onde parziali, per diffusione da potenziale, quando le variabilik (numero d’onda) eλ=l+1/2 sono simultaneamente complesse. Alcuni risultati sulla posizione dei poli della matriceS derivano da un procedimento di completamento analitico. Le proprietà qui derivate della matriceS sono esattamente equivalenti alle formule di dispersione doppie dell’ampiezza di diffusione nell’energia e nel momento trasferito.
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References
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This had not been achieved in (1,2) because there we assumed from the beginning the validity of ordinary dispersion relations, as it follows from the earlier work ofKhuri and others (3).
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This is identity (1.8) of paper (1). The functionsC(λ) andS(λ) are linear combinations of Jost functions.
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In (1) we used the notations=zp. The formula we gave there forJ(z) is misprinted and the reader should refer to this paper in order to avoid this and other misprints which unfortunately occurred in (1).
In (3)R. Blankenbecler, M. L. Goldberger, N. N. Khuri andS. B. Treiman:Ann. of Phys.,10, 62 (1960)
Recently we have found a paper (8) where the name « spiralling states » has been adopted.
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A more complete investigation on this subject has been carried out in a forthcoming paper by two of us (A.B.-A.M.L.): Holomorphy domain of theS-matrix in potential scattering (submitted toNuovo Cimento).
We are indebted to Prof.R. Ascoli for very interesting discussions on this matter and to Prof.V. Glaser for his kind permission to use his results in this paper.
An accurate discussion about poles in the case of complex values ofk (λ real) has been carried out byR. G. Newton (9).
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This property off(E, cosθ) is true only for potentials which are of the Yukawa type. In (2–4) are reported proofs which are essentially based on the perturbative expansion of the scattering amplitude in momentum space. In (1) this analyticity is derived with the help of the present W.K.B. approximation and of the Watson transform (2.13). This proof is the one reported here.
For a complete discussion of this point, see Lectures deliverel by one of us (T.R.) at the Summer School on High Energy Physics, in Hercegnovi, Yugoslavia (1961).
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Bottino, A., Longoni, A.M. & Regge, T. Potential scattering for complex energy and angular momentum. Nuovo Cim 23, 954–1004 (1962). https://doi.org/10.1007/BF02731254
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DOI: https://doi.org/10.1007/BF02731254