Summary
We examine the problem of reconstructing, at a fixed energy, the amplitude of an inelastic reactionAB → CD from its differential cross-section. We furthermore assume that the inelasticity proceeds from the coupling to two-body channels and that we know entirelyone of the two elastic amplitudesAB → AB, CD → CD, but not both. We first prove that, in the case of an infinite number of partial waves, the trivial ambiguities of the reconstructed amplitudes are reduced to a possible complex conjugation eventually combined with a change of sign. Secondly we study the particular case of inelastic amplitudes described with a finite number of partial waves. ForL max ≤ 5 it is shown that no more than three nontrivially related solutions can exist simultaneously.
Riassunto
Si esamina il problema di ricostruire, ad un’energia fissata, l’ampiezza di una reazione anelasticaAB → CD dalla sua sezione d’urto differenziale. Si suppone inoltre che l’anelasticità provenga dall’accoppiamento a canali di due corpi e che si conosca completamenteuna delle due ampiezze elasticheAB → AB, CD → CD, ma non entrambe. Si dimostra dapprima che nel caso di un numero infinito di onde parziali le ambiguità banali delle ampiezze ricostruite si riducono ad una possibile coniugazione complessa eventualmente combinata con un cambio di segno. Poi si studia il caso particolare di ampiezze anelastiche descritte con un numero finito di onde parziali. PerL max ≤ 5 si mostra che non possono esistere simultaneamente più di tre soluzioni non banalmente correlate.
Реэюме
Мы исследуем проблему реконструкции при фиксированной знергии амплитуды неупругой реакции АВ → CD иэ дифференциального поперечного сечения зтой реакции. Кроме того, мы предполагаем, что неупругость следует иэ свяэи с двух-частичными каналами и что мы полностью энаем только одну иэ двух упругих амплитуд АВ → АВ, ОD → СD. Сначала мы докаэываем, что в случае бесконечного числа парциальных волн тривиальные неодноэначности реконструированных амплитуд сводятся к воэможному комплексному сопряжению, свяэанному в конечном счете с иэменением энака. Затем мы рассматриваем частный случай неупругих амплитуд, описываемых с помошью конечного числа парциальных волн. Для Lmax≤ 5 покаэывается, что одновременно может сушествовать не более трех нетривиальных соотносительных рещений.
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References
This paper constitutes a revised and amplified version of a preliminary work of one of us:M. Benayoun: CERN preprint TH. 1960 (1974) (not to be published).
H. Cornille andJ. M. Drouffe:Nuovo Cimento,20 A, 401 (1974).
C. Itzykson andA. Martin:Nuovo Cimento,17 A, 245 (1973).
For general reviews of the subject, see, for instance,A. Martin: CERN preprint TH. 1764 (1973);H. Burkhardt: CERN preprint TH. 1784 (1973);H. Cornille: I.P.N.O. TH. 74-25 (1974);M. de Roo: Thesis (1974).
M. Benayoun: Thesis (Paris, 1973) (unpublished).
R. F. Alvarez-Estrada, B. Carreras andG. Mahoux: Saclay preprint DPhT/74/35 (1974).
In a preliminary stage of the study,A. Martin and one of us (M. Benayoun) obtained results in the polynomial case up toL=3.
We thank very muchA. Martin for his very important contributions to the results presented in this Section.
J. Bros, H. Epstein andV. Glaser:Nuovo Cimento,31, 1265 (1964);Comm. Math. Phys.,1, 240 (1965).
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Part of this work has been done at the Theory Division at CERN, which has provided kind hospitality.
The computations were performed by the computer of the University of Thessaloniki.
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Benayoun, M., Cornille, H. On the reconstruction of coupled two-body channel amplitudes. Nuov Cim A 30, 1–33 (1975). https://doi.org/10.1007/BF02729790
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DOI: https://doi.org/10.1007/BF02729790