Meson-Meson Interaction in the Bethe-Salpeter Approximation

The $\pi -\pi$ interaction is studied in the Bethe-Salpeter approximation by first obtaining the "effective" interaction Hamiltonian with the help of the lowest order $S$ matrix for $\pi -\pi$ scattering. This Hamiltonian is then used to set up a Goldberger-type integral equation for $\pi -\pi$ scattering which is solved for the cases of total isotopic spins $I=0, 2$, assuming the mesons to be in $s$ states with respect to each other. It is found that the interaction in the state $I=0\mathrm{}$ is too strongly attractive to give a resonance. On the other hand, a resonance is obtained for the case $I=2\mathrm{}$, at a momentum (in the center-of-mass system) $k\sim 93$ Mev/c for each meson, taking $\frac{G^2}{4\pi }=15.5\mathrm{}$.