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A new treatment of the left-hand cut for N / D problems is presented whereby, instead of poles, the cut is parametrized by a Legendre expansion which fixes the discontinuity. The method is tested through the use of analytic forms, and is found to work very well in a wide variety of circumstances. An extension is made to situations where there are errors in the input, and resulting problems are discussed. The method is applied to elastic pion-pion scattering. Both the rho and f 0 mesons are produced dynamically, but trouble is encountered for the s-waves. The relation of this trouble to ambiguities found in other s-wave calculations is discussed.

Restrictions on low-energy s-wave behavior of pion-pion scattering are studied using the N / D method and the dynamical assumption of no Castillejo-Dalitz-Dyson poles. Within the context of a model corresponding to rather well determined phenomenology, we find rather tight restrictions on the low-energy constants.

Using only analyticity, crossing symmetry, and unitarity, we find two distinct types of pion-pion scattering amplitudes. One class includes a wide range of non-physical solutions of the “s-wave dominant” type; these can be rejected only through assumptions other than crossing symmetry and unitarity. The second class, also found by other authors, contains the physical amplitudes. Our solutions, plus some further experimental information, agree well with the scattering lengths suggested by Morgan and Shaw, α 0 = 0.16 ± 0.04, α 2 = −0.05 ± 0.01.