Pion-Nucleus Scattering Lengths

The soft-pion theory and the Fubini-Furlan mass dispersion relations have been used to analyze the pion-nucleon scattering lengths and obtain a value for the $\sigma$ commutator term. With this value and using the same principles, scattering lengths have been predicted for nuclei with mass number ranging from 6 to 23. Agreement with experiment is very good. For those who believe in the Gell-Mann-Levy $\sigma$ model, the evaluation of the commutator yields the value $0.26{\left(\frac{m_{\sigma }}{m_{\pi }}\right)}^2$ for the $\sigma$-nucleon coupling constant. The large dispersive corrections for the isosymmetric case imply that the basic idea behind many of the soft-pion calculations, namely, slow variation of matrix elements from the soft-pion limit to the physical pion mass, is not correct.