Analysis of Pion-Helium Scattering for the Pion Charge Form Factor

Elastic scattering of ${\pi }^{+}$ and ${\pi }^{-}$ on ${\mathrm{He}}^4$ is analyzed for information on the charge radius of the pion using a new method based on boundary conditions near the nuclear surface. The pion radius enters the calculation via the electrostatic potential of the pion and helium charge distributions, which is assumed to be the only charge-dependent interaction. Since ${\mathrm{He}}^4$ is isoscalar, the strong nuclear interaction is assumed to be charge-independent. Differential cross-section data for both signs of the charge are fitted simultaneously by a program that uses the logarithmic derivatives of the pion radial wave function for each charge as free parameters. If the nuclear interaction operator is symmetric (i.e., $〈{\overrightarrow{\mathrm{r}}}^{\prime }\left|U_N\right|\overrightarrow{\mathrm{r}}〉=〈\overrightarrow{\mathrm{r}}\left|U_N\right|{\overrightarrow{\mathrm{r}}}^{\prime }〉$), the difference in the logarithmic derivative for a given partial wave resulting from changing the sign of the charge may be expressed as an integral of the internal Coulomb potential weighted by the wave function. Nuclear model dependence is greatly reduced by the constraints imposed by the empirical boundary conditions on the internal wave function. The Berkeley data at beam momenta of 130 to 163 MeV/c are analyzed by this method using both local potential and Kisslinger models for the strong interaction, and Gaussian and Yukawa pion charge distributions. The results indicate $2.2<\mathrm{}{\mathcal{r}}_{\pi }<3.2\mathrm{}$ F, depending on the theoretical model, with an experimental precision of ∼ ±0.5 F. In the course of the analysis, the singular-point difficulty with the Kisslinger model was examined and found to be serious.