We treat the response in a finite nucleus to spin-isospin sensitive probes in the transverse spin channel. Since linear momentum is here not a good quantum number, we adopt a formalism suggested by Toki and Weise, which incorporates an expansion in partial waves of good nuclear total angular momentum. The pion self-energy $\Pi$ is nonlocal in momentum space; iterations of this quantity give the response function $R$, which is obtained by solving a matrix integral equation exactly (using numerical methods). We use $R$ to renormalize the matrix element of a transverse spinisospin probe and find large effects near the critical momentum region ($q\sim 2-3m_{\pi }$) and almost no effects for a small momentum transfer. We also find very important effects of the nonlocality in momentum space. We apply our formalism to the level with $J^P=1^{+}$; $T=1\mathrm{}$ in ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}$. The results are compared with currently used approximations: the local density approximation, the infinite nuclear matter approximation, and the approximations of Toki and Weise. All these approximations fail to reproduce the exact results, especially for $q\to 0$ and near the critical momentum region. Using the infinite nuclear matter approximation, we find the best agreement for the equivalent constant density $\overline{\rho }=0.08\mathrm{}$ ${\mathrm{fm}}^{-3\mathrm{}}$ for low $q$; this agreement is not found, however, for any particular value of $\overline{\rho }$ when the value of $q$ is either tiny or big. The approximation of Toki and Weise cannot be reliably used in our case, except possibly for $q\approx m_{\pi }$. The finite-nucleus results obtained are compared with the corresponding longitudinal response. The great similarity of the last two quantities, in contrast with Fermi gas estimates, is partially supported by new experimental data.

• Received 27 December 1983

DOI:https://doi.org/10.1103/PhysRevC.30.1238