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Theory of low-energy scattering in field theory

Annals of Physics
Publication Date
DOI: 10.1016/0003-4916(60)90002-6


Abstract An approximate representation for the scattering amplitude of processes with two final particles, expected to be valid below the threshold for anelastic production of new particles, is deduced from the Mandelstam double integral representation. The principle underlying this reduction is that only the contributions from the singularities of the scattering amplitude near the physically interesting range of the variables need to be taken correctly into account, while the contributions from distant singularities can be expressed in the form of polynomial expansions. This means that anelastic processes are neglected in the unitarity condition of the S matrix, and their effects lumped into arbitrary constants. The method leads to systems of coupled nonlinear integral equations for the lowest angular momentum partial waves of the different reactions described by the same amplitude. These equations coincide, for the problem of pion-pion scattering, with those found by Chew and Mandelstam with a different procedure. Meson-nucleon and nucleon-nucleon scattering are then investigated by means of a model in which all particles are neutral and spinless. In the first problem one obtains a generalization of the equations found by means of fixed momentum transfer dispersion relations, in which the effect of a possible strong pion-pion interaction is included. For nucleon-nucleon scattering the contribution of the two meson exchange to the scattering amplitude is expressed in terms of pion-pion and pion-nucleon amplitudes. This should be sufficient for the determination of most of the nucleon-nucleon phaseshifts, apart possibly from a few parameters (scattering lengths and effective ranges) of the lowest ones which depend critically on the higher mass states. The possibility of constructing a two-meson exchange static potential is also demonstrated.

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