Solvable model for pion-pionSandPwaves derived from noncovariant perturbation theory

Motivated by Lagrangian field theory, we construct an effective Hamilton operator for $P$- and $S$-wave pion-pion interactions, which can be handled easily by noncovariant perturbation methods. The result is a factorable $T$-matrix equation with unambiguous off-shell behavior, which includes the effects of intermediate states with $\epsilon \mathrm{}\left(1300\right)\mathrm{}$, $S^{*}\mathrm{}\left(980\right)\mathrm{}$, and $\rho \mathrm{}\left(770\right)\mathrm{}$ mesons as well as with $K\overline{K}$ and $N\overline{N}$ pairs and describes the influence of inelastic pair effects on elastic scattering. The numerical calculations are in good agreement with the experimental $S_{00}$ and $P_{11}$ amplitudes up to a scattering energy of 2.5 GeV; a range of fundamental importance for nuclear physics.