Pion-pion scattering lengths from Veneziano formula and η→3π,K→3π decays

Abstract

Theη→ 3π decay amplitude is factorized into (a) the transition matrix element fromη toπ ⁰ and (b) Lovelace's expression of the Veneziano type for the pion-pion scattering amplitude. (a) is related with the electromagnetic mass splitting betweenK ⁺ andK ⁰. The absolute value of (b) is then obtained by fitting theη decay width. The pion-pion scattering lengths from (b) are reasonable. Since the analogous treatment ofK→ 3π decay together with a soft-pion determination of theK→π transition matrix element yield very small scattering lengths we suggest that theK→2π transition matrix element depends strongly on the pion masses.