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We investigate the relation between different three-dimensional reductions of the Bethe-Salpeter equation and the analytic structure of the resultant amplitudes in the energy plane. This correlation is studied for both the φ²σ interaction Lagrangian and the πN system with s-, u-, and t-channel pole diagrams as driving terms. We observe that the equal-time equation, which includes some of the three-body unitarity cuts, gives the best agreement with the Bethe-Salpeter result. This is followed by other three-dimensional approximations that have less of the analytic structure.
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Lahiff et al. (2002) studied this question.
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