Singularities in the X-Ray Absorption and Emission of Metals. II. Self-Consistent Treatment of Divergences

The many-body approach of the preceding paper is generalized to include self-energy and vertex renormalization is self-consistent fashion. The algebra of parquet diagrams is developed in terms of an unknown divergent parameter (related to the deep-hole Green's function $\mathcal{G}$, and of an unknown irreducible interaction $R$. The resulting interaction operator $\gamma$ is used to calculate $\mathcal{G}$ and $R$; the corresponding coupled equations are solved self-consistently in the weak coupling limit. The spectral densities of the deep-hole Green's function and of the x-ray response function are calculated explicitly; in the weak coupling limit, the Mahan singularity found in the preceding paper is unaffected by renormalization. The self-consistent formalism presented here describes the reaction of divergent fluctuations on themselves, and should, therefore, be useful in other more complicated problems, such as the Kondo effect.