Il Nuovo Cimento A (1965-1970)

, Volume 64, Issue 1, pp 225–242 | Cite as

Constraints on ππ partial waves from positivity and analyticity

  • F. J. Yndurain


We consider the constraints on the partial waves for π0π0 or ππ in the isospin-zero channel, that follow from analyticity and positivity. We show that, if these two requirements are met, then a set of inequalities on the partial waves withl≧2 and energy below threshold must be satisfied. The converse is also proved showing that, if these constraints are fulfilled, then an analytic scattering amplitude with positive absorptive part can be constructed. The practical importance of these conditions is discussed showing that, for example, they imply the inequalities (pion mass=1)al+2<(1/16)((l+2)(l+1)/(l+5/2)(l+3/2))al; hereal are the scattering lengths,al=limfl(s)/q2l. Furthermore, these inequalities (which had essentially been also found by Martin) are saturated very accurately whenl≧4. The relevance of our analysis for models of ππ scattering is discussed.


Partial Wave Moment Problem Pion Mass Absorptive Part Prova 
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Ограничения на пп парцнальные волны из полжительности и аналитичности


Мы рассматриваем ограничения на парциальные волны для π0π0 или ππ, в канале с нулевым изоспином, которые следуют из аналитичности и положительности. Мы показываем, что если встречаются эти два требования, то тогда должна удовлетворяться система неравенств на парциальные волны сl≧2 и энергией ниже порога. Также доказывается и обратное, покзываюЩее, что если эти ограничения выполняются, то тогда может быть сконструирована амплитуда рассеяния с положительной абсорбционной частью. Обсуждается практическая важность этих условий, отмечая, что они, например, предполагают неравенства (пионная масса=1)al+2<(1/16)((l+2)(l+1)/(l+5/2)(l+3/2))al; здесьal представляют длины рассеяния,al=limfl(s)/q2l. Кроме того, эти неравенства (которые, по существу, также были найдены Мартиным) насыщаются очень аккуратно, когдаl≥4. Обсуждается уместность нашего анализа для моделей ππ рассеяния.


Si considerano i vincoli sulle onde parziali per π0π0 o ππ nel canale di isospin zero, che seguono dall'analiticità e positività. Si dimostra che, se si verificano queste due condizioni, allora si deve soddisfare un insieme di diseguaglianze sulle onde parziali conl≧2 ed energia al di sotto della soglia. Si prova anche l'inverso mostrando che, se questi vincoli sono soddisfatti, si può costruire un'ampiezza di scattering analitica con parte assorbitiva positiva. Si discute l'importanza pratica di queste condizioni mostrando che, ad esempio, esse implicano le diseguaglianze (massa del pione=1)al+2<(1/16)((l+2)(l+1)/(l+5/2)(l+3/2))al dove glial sono le lunghezze di scattering,al=limfl(s)/q2l. Inoltre queste disuguaglianze (che sono state essenzialmente trovate anche da Martin) sono saturate molto accuratamente quandol≧4. Si discute l'importanza della nostra analisi per modelli dello scattering ππ.


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Copyright information

© Società Italiana di Fisica 1969

Authors and Affiliations

  • F. J. Yndurain
    • 1
  1. 1.CERNGeneva

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