Thus, the left-hand side of Eq. (76) is identical to the right-hand side as we can see from Eq. (91). Therefore, the ST identity for the vectorial vertex is satisfied at one-loop. Writing the expansions for ˜ Γ $^{µνλ}$( p, q, k ) and ˜ G λ ν ′ ( p, q, k ), defined in (59) and (61), as we obtain the ST identity (69) for the triple photon vertex, in the tree approximation, which requires that On the other hand, the one-loop approximation is given by where we have introduced the photon self-energy Π νν $^{′}$( q ) ≡ Π$_{T}$( q $^{2}$)( q $^{2}$g νν ′ − q $^{ν}$qν $^{′}$) + Π$_{θ}$ ( q $^{2}$)˜ q $^{ν}$˜ q ν $^{′}$. The contributions with a fermion loop in the left and right-hand sides of Eq. (103) are directly identified when we consider the diagrams: − ∫ d $^{4}$l (2 π ) 4 i ( p + q − l ) 2 i ( p − l ) $_{2}$( ie )( / p − / l ) iS$_{0}$ ( l )( ie )( / p − / l ) ( 1 − e 2 il ∧ $^{q}$e − 2 ip ∧ q ) + i d $^{4}$l (2 π ) 4 i ( p − l ) 2 i ( p + q − l ) $_{2}$( ie ) / q ( ie ) ( 1 − e 2 il ∧ $^{q}$e − 2 ip ∧ q ) . (98) ˜ Γ $^{µνλ}$( p, q, k ) = 2 eZ$_{1}$ sin( p ∧ q ) γ $^{µνλ}$( p, q, k ) + 2 e sin( p ∧ q )Λ $^{µνλ}$( p, q, k ) , (99) ˜ G λ ν ′ ( p, q, k ) = − 2 e ˜ Z$_{1}$ sin( p ∧ q ) g λ ν ′ + 2 e sin( p ∧ q ) B λ ν ′ ( p, q, k ) , (100) Z$_{1}$ ˜ Z$_{3}$ [ ( k $^{2}$g λν − k $^{λ}$k$^{ν}$) − ( q $^{2}$g νλ − q $^{ν}$q$^{λ}$) ] = ˜ Z$_{1}$Z$_{3}$ [ ( k $^{2}$g λν − k $^{λ}$k$^{ν}$) − ( q $^{2}$g νλ − q $^{ν}$q$^{λ}$) ] , (101) ˜ Z$_{3}$/ ˜ Z$_{1}$ = Z$_{3}$/Z$_{1}$. (102) p$_{µ}$γ $^{µνλ}$( p, q, k ) b ( p $^{2}$) + p$_{µ}$ Λ $^{µνλ}$( p, q, k ) = Π $^{νλ}$( q ) − Π $^{λν}$( k ) + 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 $^{′}$) , (103) = 2 e sin( p ∧ q )Λ µνλ a 1 ( p, q, k ) (104) = − ∫ d $^{4}$l (2 π ) $_{4}$tr( − ieγ $^{ν}$) iS$_{0}$ ( q + l )( − ieγ $^{µ}$) iS$_{0}$ ( p + q + l ) B. The triple photon vertex 16