content_image/346413.jpg content_image/346417.jpg content_image/346418.jpg content_image/346419.jpg content_image/346420.jpg
diagrams is given by 

Here P and Q are the permutation operators, interchanging the valence ( a ) and the core ( b ) electron states, ( − 1) P is the sign of the permutation P . The summation over b runs over two core electron states with different projections of the total angular momentum. In what follows, we will also use the notation Δ ≡ ε$_{Qb}$ − ε$_{Pb}$ . 

<img src='content_image/346413.jpg'>

One has to pay special attention to the ultraviolet (UV) divergencies, that arise in the formal expressions under consideration. First, we introduce the unrenormalized self-energy operator, 

Here and below the integration over ω is carried out from −∞ to + ∞ . Every diagram involving the self-energy loop has to be considered together with the corresponding diagram with the mass counterterm, what results in the replacement Σ( ε ) → Σ$_{R}$( ε ) = Σ( ε ) − γ $^{0}$δm . The matrix elements of Σ$_{R}$( ε ) still have the divergent part: 

<img src='content_image/346417.jpg'>

<img src='content_image/346418.jpg'>

where B (1) is the UV-divergent constant. Assuming that | p 〉 and | q 〉 obey the Dirac equation (1) we have, 

<img src='content_image/346419.jpg'>

In order to isolate the divergent part we follow the potential-expansion approach [50]. The fi- nite part of the self-energy matrix element is then divided into the zero- and one-potential terms evaluated in momentum space, and the many-potential term evaluated in coordinate space. The diagrams with one vertex inside the self-energy loop also suffer from UV-divergencies. It can be shown that for an arbitrary operator U , 

<img src='content_image/346420.jpg'>

In our case U is either T$_{0}$ or I (Δ) (in the latter case 〈 p | U | q 〉 ≡ 〈 pr | I (Δ) | qs 〉 ). Due to the Ward identity we have L (1) = − B $^{(1)}$. The finite part of the vertex contributions is divided into t he zero- potential term evaluated in momentum space and the many-potential term evaluated in coordinate space. 

