Summary
Kinoshita, Loeffel andMartin have obtained recently for spinless particles bounds for high-energy scattering at fixed angle. For this, they imposed some of the constraints required by unitarity and analyticity. The method consisted of substituting a power series in place of the usual Legendre polynomial expansion of the scattering amplitude. It is the purpose of this work to extend these bounds for the scattering of particles with spin 1/2, 1/2: nucleon-nucleon scattering. Here it is found that the extension of results to nucleon-nucleon scattering is made by replacing Legendre polynomials by Jacobi polynomials, taking advantage of the great similarity of their properties. The main results are: 1) for forward scattering (with θ≠0), for four of the five linearly independent helicity amplitudes, we find the same energy and angle-dependence for the bounds as in the spinless case ϕi(s, cos θ) | < const s-1/2(log s)3/2/(sin θ)2 ; the fifth giving a slightly supplementary term; 2) for backward scattering (with θ≠π a symmetric situation; 3) at fixed physical momentum transfer ≠0 we find the same energy-dependence as in the spinless case.
Riassunto
Kinoshita, Loeffel e Martin hanno di récente ottenuto limiti per le particelle senza spin nello scattering di alta energia ad angolo fisso. A questo scopo essi imposero alcune delle restrizioni richieste dairunitarietà e daU’analiticità. Il metodo consisteva nel sostituire una série di potenze al posto del solito sviluppo dell’ampiezza di scattering in polinomi di Legendre. È scopo di questo lavoro estendere questi limiti allo scattering di particelle con spin 1/2,1/2 scattering nucleone-nucleone. Qui si trova che l’estensione dei risultati allo scattering nucleone-nucleone si puö fare sostituendo i polinomi di Legendre con polinomi di Jacobi, traendo vantaggio délia grande rassomiglianza delle loro proprietà. I principali risultati sono : 1) per scattering in avanti (con 0 #0), per quattro delle cinque ampiezze di elicità linearmente indipendenti troviamo la stessa dipendenza daU’energia e dall’angolo per i limiti come nel caso senza spins, cos 0)]< < const s~=(logs)t/(sin 0)2, mentre la quinta dà un piccolo termine supplementäre, 2) per scattering all’indietro (condy^n) una situazione simmetrica e 3) a momento trasferito fisso =^0 troviamo la stessa dipendenza dell’energia come nel caso senza spin.
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Cornille, H. Bound for the high-energy nucleon-nucleon scattering amplitude at angle ≠0 or π and at fixed momentum tranfer ≠0. Nuovo Cim 31, 1101–1115 (1964). https://doi.org/10.1007/BF02821682
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DOI: https://doi.org/10.1007/BF02821682