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We have evaluated the 71 ~ scattering lengths, using current algebra but without the use of a power series expansion or extrapolation of the scatter-ing amplitude. We have used the usual LSZ reduction and PCAC in terms of II an axial current J = A + cB $ where A is the usual axial current, c is the Goldberger-Treiman constant, and Q, is the pion field. The scattering am-plitude is decomposed into four terms: Two of the three terms which are due to the equal-time commutators, are evaluated by assuming that the usual cur- ’ rent algebra of the current A holds for the current J, Other terms are evalu-I ated in terms of the single-particle intermediate states, among which we show that only s-waves contribute at threshold, Assuming that the E-resonance is the only s-wave dominating the low-energy (rn) scattering, we find a relation between the form factors arising from the equal-time commutators of the current J, and the E-pion coupling constant. Finally we obtain the scattering lengths, corresponding to isospins 0, and 2 and E-resonance width 200 MeV,-1-1 as a o = 0.278 m and a2 =-0.044 m

Year: 1970

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