Energy Dependence of the Nucleon-Nucleon Phase Shifts

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Abstract
Starting from the analytic structure of partial wave amplitudes predicted by the Mandelstam representation, relativistic formulas are derived for the energy dependence of the phase shifts for nucleon-nucleon scattering, neglecting inelastic processes. These formulas depend on integrals over functions defined by a (numerically) soluble integral equation whose kernel is determined from the absorptive part of the amplitude in the nonphysical region. The contribution to this kernel from single-pion exchange is explicitly exhibited and the contribution from two-pion exchange is calculable. A generalization of the formulas to include phenomenological constants representing the unknown contribution of multimeson and other particle exchanges is given. The dependence of the phase shifts on these parameters is sufficiently simple to allow the formulas to be used for the least-squares fitting of empirical data. Further, these constants can be varied independently, and as much empirical information as is desired can be incorporated into the formulas without destroying this independence. In the case of coupled states, the phenomenological formulas satisfy unitarity only approximately; this approximation can be removed by a subsidiary calculation, which destroys the independence of the parameters for these states. Because of the neglect of inelastic processes, the range of validity of the formulas is expected to be from 0 to approximately 400 Mev.