Theory ofs−dScattering in Dilute Magnetic Alloys. II. Derivation and Solution of Linear Vertex Equations

A set of linear equations for the vertex function associated with conduction-electron scattering from a spin-dependent impurity potential is derived from perturbation theory and is shown to lead to the Abrikosov vertex function in the limit of a contact interaction and logarithmic accuracy. The general reduction to quadratures of the linear equations is given for $s$-wave, factorizable interactions. A detailed evaluation of the (off-energy-shell) vertex function is performed for that interaction which corresponds to the pole-approximation solution to the $s$-wave Low equation in the limit of spin-independent potentials. The Fermi factors introduced by a spin-dependent $s-d$ interaction lead to a (linear) vertex function with analytical properties differing from those of the pole-approximation solution to the coupled Low equations. The influence of ground-state correlations in the linear equations causes the failure of inelastic, single-channel unitarity in both spin channels ($J=S\pm \frac{1}{2}$) for factorizable, $s$-wave antiferromagnetic $s-d$ interactions with any strength or form factor.