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Solving the four-dimensional $NN-πNN$ equations for scalars below meson-production threshold

The four-dimensional $NN-πNN$ equations are adapted to the case of scalar particles with a $ϕ^2σ$ interaction Lagrangian, and solved for energies below the $σ$-production threshold. This is achieved in the approximation where $ϕσ$ scattering is dominated by the $s$-channel $ϕ$-pole term. The importance of the removal of double-counting is investigated, and a detailed comparison of the results of a covariant coupled-channels formulation and the Bethe-Salpeter equation in the ladder and ladder plus crossed-box approximations is presented. A brief discussion of the extension of the method to energies above the $σ$-production threshold is given.