where their PP contributions are From this, we see that the renormalization constant ˜ Z$_{1}$ must be so that ˜ Z$_{1}$ = Z$_{5}$ and thus we obtain the relations Therefore, the sum of all PP contributions of the right-hand side of Eq. (103) becomes which is the same result as for the left-hand side, Eq. (120). Besides these ultraviolet divergent parts, arising from the planar parts of the diagrams, we have also infrared singular parts (SP) coming from the nonplanar parts of the same diagrams, at p, q, k = 0. Explicit calculations, combining denominators with Feynman parameters and using nonplanar integrals, give us the SP for the diagrams Λ µνλ on the left-hand side of Eq. (103): where we are not taking into account the logarithmic singularities. Contracting q$_{µ}$ in the above expression, we obtain which is exactly the same SP for the photon self-energy diagrams on the right-hand side of Eq. (103), The other diagrams of the ST identity (103) contribute only with logarithmic SP. These singularities are not problematic as they are integrable. = i ( − i ) $^{2}$(2 e ) 3 ∫ d $^{4}$l (2 π ) 4 ( − l )$_{α}$ γ αν λ ′ ( p + l, q, − p − q − l ) l $^{2}$( p + l ) $^{2}$( p + q + l ) 2 × sin( l ∧ p ) sin( − l ∧ p − l ∧ q ) sin( l ∧ q + p ∧ q ) , PP[ B λ ν ′ ( p, q, k )( q $^{2}$g νν ′ − q $^{ν}$qν $^{′}$) + B ν λ ′ ( p, k, q )( k $^{2}$g λλ ′ − k $^{λ}$kλ $^{′}$)] = − e 2 16 π 2 1 ϵ [( k $^{2}$g λν − k $^{λ}$k$^{ν}$) − ( q $^{2}$g νλ − q $^{ν}$q$^{λ}$)] . (129) ˜ Z$_{1}$ = 1 − e 2 16 π 2 1 ϵ , (130) Z$_{2}$/Z$_{1}$$_{F}$ = ˜ Z$_{3}$/ ˜ Z$_{1}$ = Z$_{3}$/Z$_{1}$. (131) P P [Π νλ $_{b,c,d}$( q ) − Π λν $_{b,c,d}$( k ) + B λ ν ′ ( p, q, k )( q $^{2}$g νν ′ − q $^{ν}$qν $^{′}$) + B ν λ ′ ( p, q, k )( k $^{2}$g λλ ′ − q $^{λ}$qλ $^{′}$)] = 7 e 2 48 π 2 1 ϵ [( k $^{2}$g λν − k $^{λ}$k$^{ν}$) − ( q $^{2}$g νλ − q $^{ν}$q$^{λ}$)] , (132) SP[2 e sin( p ∧ q )Λ µνλ b 1 ,b 2 ,c,d 1 ,d 2 ,d $_{3}$( p, q, k )] = 4 e 3 π 2 sin( p ∧ q ) p ∧ q ( ˜ p $^{µ}$˜ p $^{ν}$˜ p λ ξ ˜ p 4 + ˜ q $^{µ}$˜ q $^{ν}$˜ q λ ξ ˜ q 4 + ˜ k µ ˜ k ν ˜ k λ ξ ˜ k 4 ) , (133) SP[ q$_{µ}$ Λ µνλ b 1 ,b 2 ,c,d 1 ,d 2 ,d $_{3}$( p, q, k )] = 2 e 2 π 2 ( ˜ q $^{ν}$˜ q λ ξ $^{2}$˜ q 4 − ˜ k ν ˜ k λ ξ 2 ˜ k 4 ) , (134) SP[Π νλ $_{b,c,d}$( q ) − Π λν $_{b,c,d}$( k )] = 2 e 2 π 2 ( ˜ q $^{ν}$˜ q λ ξ $^{2}$˜ q 4 − ˜ k ν ˜ k λ ξ 2 ˜ k 4 ) . (135) 22