Second-order pion-nucleus optical potential

The second-order pion-nucleus optical potential is expressed in momentum space in terms of the $\pi N$ off-shell $t$ matrix and the $\mathrm{NN}$ correlation function. The dependence of the $\pi N t$ matrix on the total $\pi N$ momentum is determined within the framework of relativistic particle quantum mechanics. Numerical results are presented for $\pi -{}_{}{}^4{}_{}{}^{}\mathrm{He}$ scattering. It is found that the $\mathrm{NN}$ correlations tend to decrease the forward elastic cross sections by a factor of about 1.5 at low energy, $E_{\pi }\simeq 60$ MeV. At higher energies the main effect of the $\mathrm{NN}$ correlations is to increase the differential cross section at large angles. The effect of $\mathrm{NN}$ correlations on the total cross section and the pion wave function is also found to be important. A study of the dependence of the calculated cross sections on the range parameters of the $\pi N t$ matrix and of the $\mathrm{NN}$ correlations suggests that the $\mathrm{low}$-energy pion-nucleus scattering data cannot be adequately described by an optical model constructed from the first- and second-order terms in an expansion in terms of the $\pi N t$ matrix.