in which the self-energy has been expressed as i Σ$_{C}$( q ) = q $^{2}$b ( q $^{2}$). Thus, with the expressions (31) and (46), we can rewrite the identity (42) as follows By considering that energy-momentum conservation holds at the vertices Γ $^{µ}$( k, p, q ) and H ( k, p, q ), we can write and With this representation we may obtain from Eq. (47) that To obtain ST identity for the triple photon vertex, the proper part of 〈 0 | T ( A$_{µ}$A$_{ν}$A$_{λ}$ ) | 0 〉 , we differentiate the functional equation (15) with respect to ζ ( x ), J $^{ν}$( y ) and J $^{λ}$( z ) and turn off all the sources. The result is or in terms of the Green functions, where D AD ν ( y ) denotes the covariant derivative in the adjoint representation, D AD ν ( y ) C ( y ) = ∂$_{yν}$C ( y ) − ie [ A$_{ν}$ ( y ) , C ( y )]$_{⋆}$. Thus, we can rewrite the above expression as where, after the application of the differential operators, we must identify ˆ y and ˆ z respectively with y and z . q$_{µ}$ Γ $^{µ}$( k, p, q )[1 + b ( q $^{2}$)] = ie [ S − 1 F ( k ) H$_{1}$ ( k, p, q ) − H$_{2}$ ( k, p, q ) S − 1 F ( p )] . (47) Γ $^{µ}$( k, p, q ) = ie (2 π ) $^{4}$δ$^{4}$( k − p − q ) ˜ Γ $^{µ}$( p, p + q ) (48) H ( k, p, q ) = (2 π ) $^{4}$δ$^{4}$( k − p − q ) ˜ H ( p, p + q ) . (49) q$_{µ}$ ˜ Γ $^{µ}$( p, p + q )[1 + b ( q $^{2}$)] = S − 1 F ( p + q ) ˜ H$_{1}$ ( p, p + q ) − ˜ H$_{2}$ ( p, p + q ) S − 1 F ( p ) . (50) 1 αe ∂ µ x δ $^{3}$W δJ $^{µ}$( x ) δJ $^{ν}$( y ) δJ $^{λ}$( z ) ∣ = δ $^{3}$W δζ ( x ) δK $^{ν}$( y ) δJ $^{λ}$( z ) ∣ + δ $^{3}$W δζ ( x ) δJ $^{ν}$( y ) δK $^{λ}$( z ) ∣ , (51) − 1 α ∂ µ x 〈 0 | T ( A$_{µ}$ ( x ) A$_{ν}$ ( y ) A$_{λ}$ ( z )) | 0 〉 = 〈 0 | T ( ¯ C ( x ) D AD ν ( y ) C ( y ) A$_{λ}$ ( z )) | 0 〉 + 〈 0 | T ( ¯ C ( x ) A$_{ν}$ ( y ) D AD λ ( z ) C ( z )) | 0 〉 , (52) − 1 α ∂ µ x 〈 0 | T ( A$_{µ}$ ( x ) A$_{ν}$ ( y ) A$_{λ}$ ( z )) | 0 〉 = ∂$_{yν}$ 〈 0 | T ( ¯ C ( x ) C ( y ) A$_{λ}$ ( z )) | 0 〉 +2 e sin( ∂$_{y}$ ∧ ∂$_{ˆ}$ $_{y}$) 〈 0 | T ( ¯ C ( x ) A$_{ν}$ ( y ) C (ˆ y ) A$_{λ}$ ( z )) | 0 〉 + ∂$_{zλ}$ 〈 0 | T ( ¯ C ( x ) A$_{ν}$ ( y ) C ( z )) | 0 〉 +2 e sin( ∂$_{z}$ ∧ ∂$_{ˆ}$ $_{z}$) 〈 0 | T ( ¯ C ( x ) A$_{ν}$ ( y ) A$_{λ}$ ( z ) C (ˆ z )) | 0 〉 , (53) 9 C. The triple photon vertex