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Some bounds on pion-pion scattering amplitudes and their applications

Некоторые границы для амплитуд пион-пионного рассеяния и их применения

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Il Nuovo Cimento A (1965-1970)

Summary

If one uses the method introduced by Meiman and further developed by Martin, together with unitarity, crossing symmetry and analyticity, upper and lower bounds to the π0π0 scattering amplitudes inside the Mandelstam triangle are obtained in terms of the scattering lengths. These bounds are used to set a lower bound to theS-wave scattering lengtha 0 and also upper and lower bounds to the renormalized pion coupling constantλ. In terms of theD-wave scattering lengtha 2, the results area 0⩾−23.18√a 2+129.07a 2 and −7.71√a 2⩽λ⩽7.71 · √a 2+2.5a 2. Fora 2=7 · 10−4 they yielda 0⩾−0.52 and |λ|⩽0.20, where the pion mass is set equal to unity.

Riassunto

Se si usa il metodo introdotto da Meiman e successivamente ampliato da Martin, si trovano con l’unicità, la simmetria incrociata e l’analiticità, limite superiori e inferiori alle ampiezze di scattering π0π0 internamente al triangolo di Mandelstam espressi in funzione delle lunghezze di scattering. Si usano questi limiti per porre un limite inferiore alla lunghezzaa 0 di scattering dell’ondaS e anche i limiti superiori e inferiori alla costante di accoppiamento rinormalizzata del pioneλ. I risultati, in funzione delle lunghezze di scatteringa 2 dell’ondaD, sonoa 0⩾−23.18√a 2+129.07a 2 e −7.71√a 2⩽λ⩽7.71√a 2+2.5a 2. Pera 2=7 · 10−4 si trovaa 0⩾−0.52 e |λ|⩽0.20, dove la massa del pione si pone uguale all’unità.

Реэюме

Испольэуя метод, предложенный Мейманом и эатем раэвитый Мар-тиным, совместно с унитарностью, кроссинг-симметрией и аналитичностью, полу-чаются верхняя и нижняя границы для амплитуд π0π0 рассеяния внутри треугольника Манделстама, которые выражаются череэ длины рассеяния. Эти границы исполь-эуются для установления нижней границы длиныS-рассеянияa 0, а также верхней и нижней границ для перенормированной константы свяэи пионаλ. В терминах длиныD-рассеянияa 2 получаются следуюшие реэультатыa 0⩾−23.18 √a 2+129.07a 2 и 7.71 √a 2⩽λ⩽7.71 √a 2+2.5a 2. Дляa 2=7·10−4 зти соотнощения даютa 0⩾−0.52 и |λ|⩽0.20, где масса пиона считается равной единице.

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Authors and Affiliations

  1. International Centre for Theoretical Physics, Trieste, Italy

    B. K. Chung

  2. Division de Physique Théorique, Institut de Physique Nucléaire and Laboratoire de Physique Théorique des Particules Elémentaires, Université P. et M. Curie, Paris

    R. Vinh Mau

  3. Tour 16, ler ét., 4 Place Jussieu, 75230, Paris Cedex 05

    R. Vinh Mau

Authors
  1. B. K. Chung
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To speed up publication, the authors of this paper have agreed to not receive the proofs for correction.

At Division de Physique Théorique, Institut de Physique Nucléaire, during part of this work.

At the School of Mathematical and Physical Sciences, University of Sussex (England), during part of this work.

Laboratoire associé au Centre National de la Recherche Scientifique.

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Chung, B.K., Mau, R.V. Some bounds on pion-pion scattering amplitudes and their applications. Nuov Cim A 31, 193–206 (1976). https://doi.org/10.1007/BF02729726

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