Strong Pion-Alpha-Particle Scattering with Electromagnetic Corrections

Electromagnetic corrections to $\pi \alpha$ scattering for spacelike $q^2$ are calculated to see whether they can be used to determine $r_{\pi }$. The low-momentum $t$ dependence of the off-shell strong amplitude used in this calculation is determined with the help of crossing symmetry, current algebra, and the hypothesis of partially conserved axial-vector current. The relativistic corrections are found to be just as important as those which are also included in nonrelativistic treatments. When only the nonrelativistic terms are used, the large value of $r_{\pi }$ (2-3 F) obtained recently from ${\pi }^{\pm }\alpha$ scattering is reproduced. Including the relativistic terms brings the resulting value of $r_{\pi }$ down to approximately 1 F. This is consistent with the values obtained by using inelastic electron-proton scattering; however, the ${\pi }^{\pm }\alpha$ results are quite sensitive to the details of the $\pi \alpha$ nuclear interaction and to the relativistic effects, which are only roughly approximated.