Analytical theory of pion single and double charge exchange in resonance region. I. Geometrical limit

We derive analytical formulas for angular distributions $\sigma \left(\theta \right)$ of pion single and double charge exchange to analog states. The result is based on the eikonal approximation and assumes that the interactions are invariant under isospin rotations. The theory reproduces semiquantitatively the results of exact coupled channel, lowest order optical model calculations and explicates the dependence of $\sigma \left(\theta \right)$ on the structure of the nuclear target. The $\sigma \left(\theta \right)$ for single charge exchange is proportional to the square of the ratio of the valence neutron density $\Delta \rho \mathrm{}\left(R\right)\mathrm{}$ to the total density $\rho \left(\overline{R}\right)$. The theory predicts $\overline{R}$ as a function of energy and mass number $A$; for pion kinetic energy $T_{\pi }\simeq 180$ MeV, $\rho \left(\overline{R}\right)\simeq 0.1-0.2$ of central density. The $\sigma \left(\theta \right)$ for double charge exchange is proportional to ${\left[\frac{\Delta \rho \left(\overline{R}\right)}{\rho \left(\overline{R}\right)}\right]}^4$. The strong dependence of $\sigma \left(\theta \right)$ on $\frac{\Delta \rho }{\rho }$ suggests that the single and double charge exchange reactions may develop into a sensitive probe of the neutron halo. The dependence of $\sigma \left(\theta \right)$ on the diffuseness of $\rho$ and $\Delta \rho$ is also evaluated. Assuming that the neutron and proton densities are proportional, the relative $A$ dependence of $\sigma \left(\theta \right)$ for single and double charge exchange is calculated. The result is in qualitative agreement with recent measurements for single charge exchange and predicts that a similar set of measurements for double charge exchange may be feasible. Realistic densities are used to calculate the magnitude of $\sigma \left(\theta \right)$. Large enhancements are found in this case, but $\sigma \left(\theta \right)$ falls below the preliminary data by as much as a factor of 4. Difficulties in reproducing the observed angular distribution of the ${}_{}{}^{18}{}_{}{}^{}\mathrm{O}({\pi }^{+}, {\pi }^{-}){}_{}{}^{18}{}_{}{}^{}\mathrm{Ne}$ reaction are similar to those of other theories. The extent of discrepancy suggests the occurrence of strong modifications of the pion-nucleon interaction in the nuclear medium.