Δ E SE( G 2) SQED = 2 ∑ n 1 ′ 1 ε$_{Pa}$ − ε$_{n}$$_{1}$ [ 〈 P a | Σ( ε$_{Pa}$ ) | n$_{1}$ 〉〈 n$_{1}$ | T$_{0}$ | P a 〉〈 P aP b | I $^{′}$(Δ) | QaQb 〉 + 〈 P a | Σ( ε$_{Pa}$ ) | n$_{1}$ 〉〈 n$_{1}$P b | I $^{′}$(Δ) | QaQb 〉 ( 〈 Qb | T$_{0}$ | Qb 〉 − 〈 P b | T$_{0}$ | P b 〉 )] +2 〈 P a | Σ( ε$_{Pa}$ ) | P a 〉 ∑ n 1 ′ [ 〈 P aP b | I $^{′}$(Δ) | n$_{1}$Qb 〉〈 n$_{1}$ | T$_{0}$ | Qa 〉 ε$_{Qa}$ − ε$_{n}$$_{1}$ + 〈 P aP b | I $^{′}$(Δ) | Qan$_{1}$ 〉〈 n$_{1}$ | T$_{0}$ | Qb 〉 ε$_{Qb}$ − ε$_{n}$$_{1}$ ] , (28) Δ E SE( G 3) SQED = 〈 P a | Σ( ε$_{Pa}$ ) | P a 〉〈 P aP b | I $^{′′}$(Δ) | QaQb 〉 ( 〈 Qb | T$_{0}$ | Qb 〉 − 〈 P b | T$_{0}$ | P b 〉 ) . (29) Equations (19)–(29) possess ultraviolet (UV) divergences . Taking into account the mass countert- erm and employing Eq. (17) we find that Δ E SE( B ) SQED has a non-zero UV-divergent part, Δ E SE( B ) $_{SQED}$(UV) = 2 B (1) ∑ n 1 ′ 〈 P a | T$_{0}$ | n$_{1}$ 〉〈 n$_{1}$P b | I (Δ) | QaQb 〉 ε$_{Pa}$ − ε$_{n}$$_{1}$ . (30) By the end of the next subsection we will show that the sum of all the UV-divergent terms is zero. B. ”Modified vertex” diagrams For the irreducible parts of the diagrams C and F we have, Δ E SE( C ) SQED = Δ E SE( C 1) SQED + Δ E SE( C 2) SQED , (31) Δ E SE( C 1) SQED = 2 i 2 π ∫ d ω$^{ε$_{n}$$_{3}$}$$^{̸}$$^{=}$$^{ε$_{Pa}$}$ ∑ n 1 , 2 , 3 〈 P an$_{2}$ | I ( ω ) | n$_{1}$n$_{3}$ 〉〈 n$_{1}$ | T$_{0}$ | n$_{2}$ 〉 ( ε$_{Pa}$ − ω − uε$_{n}$$_{1}$ )( ε$_{Pa}$ − ω − uε$_{n}$$_{2}$ $_{)}$〈 n$_{3}$P b | I (Δ) | QaQb 〉 ( ε$_{Pa}$ − ε$_{n}$$_{3}$ ) , (32) Δ E SE( C 2) SQED = 2 i 2 π ∫ d ω$^{ε$_{n}$$_{3}$}$$^{̸}$$^{=}$$^{ε$_{Qa}$}$ ∑ n 1 , 2 , 3 〈 P an$_{2}$ | I ( ω ) | n$_{1}$n$_{3}$ 〉〈 n$_{1}$P b | I (Δ) | n$_{2}$Qb 〉 ( ε$_{Pa}$ − ω − uε$_{n}$$_{1}$ )( ε$_{Qa}$ − ω − uε$_{n}$$_{2}$ ) 〈 n$_{3}$ | T$_{0}$ | Qa 〉 ( ε$_{Qa}$ − ε$_{n}$$_{3}$ ) , (33) Δ E SE( F ) SQED = 2 i 2 π ∫ d ω$^{ε$_{n}$$_{3}$}$$^{̸}$$^{=}$$^{ε$_{Qb}$}$ ∑ n 1 , 2 , 3 〈 P an$_{2}$ | I ( ω ) | n$_{1}$Qa 〉〈 n$_{1}$P b | I (Δ) | n$_{2}$n$_{3}$ 〉 ( ε$_{Pa}$ − ω − uε$_{n}$$_{1}$ )( ε$_{Qa}$ − ω − uε$_{n}$$_{2}$ ) 〈 n$_{3}$ | T$_{0}$ | Qb 〉 ( ε$_{Qb}$ − ε$_{n}$$_{3}$ ) . (34) Since these diagrams have one vertex inside the self-energy loop, the corresponding expressions have the following structure of the ω -dependent denominators: (Δ$_{1}$ − ω ) − $^{1}$(Δ$_{2}$ − ω ) − $^{1}$. All the reducible terms that have similar structure are denoted as Δ E SE( H ) $_{SQED}$, Δ E SE( H ) SQED = Δ E SE( H 1) SQED + Δ E SE( H 2) SQED + Δ E SE( H 3) SQED , (35) 9