diagrams is given by Δ E SE SQED = ∑ b ∑ PQ ( − 1) P + Q ( Δ E SE( A ) SQED + Δ E SE( B ) SQED + Δ E SE( C ) SQED + Δ E SE( D ) SQED +Δ E SE( E ) SQED + Δ E SE( F ) SQED + Δ E SE( G ) SQED + Δ E SE( H ) SQED + Δ E SE( I ) SQED ) . (14) 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$_{Pb}$ . 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: 〈 p | Σ( ε ) | q 〉 = i 2 π ∫ d ω ∑ n 〈 pn | I ( ω ) | nq 〉 ε − ω − uε$_{n}$ . (15) 〈 p | Σ$_{R}$( ε ) | q 〉 = B $^{(1)}$〈 p | [ ε − α · p − βm − V ( x )] | q 〉 + finite part , (16) 〈 p | Σ$_{R}$( ε ) | q 〉 = B $^{(1)}$( ε − ε$_{p}$ ) δ$_{pq}$ + finite part . (17) i 2 π ∫ d ω ∑ n 1 , 2 〈 pn$_{2}$ | I ( ω ) | n$_{1}$q 〉〈 n$_{1}$ | U | n$_{2}$ 〉 ( ε − ω − uε$_{n}$$_{1}$ )( ε − ω − uε$_{n}$$_{2}$ ) = L $^{(1)}$〈 p | U | q 〉 + finite part . (18) where B (1) is the UV-divergent constant. Assuming that | p 〉 and | q 〉 obey the Dirac equation (1) we have, 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 , 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. 7