where we have used the ghost propagator (46) and the inverse of the photon propagator which satisfy Using the energy-momentum conservation at the vertices Γ$_{µνλ}$( p, q, k ) and G$_{ν}$ ′ $_{λ}$( p, q, k ), so that and we get The ST identities derived previously are valid only in a formal way since the radiative corrections contain ultraviolet divergences. To eliminate these divergences counterterms must be introduced so that the action for noncommutative QED$_{4}$ becomes where G$_{µν}$ = ∂$_{µ}$A$_{ν}$ − ∂$_{ν}$A$_{µ}$ . We begin by considering the one-loop contributions to the vectorial vertex function. Writing the ˜ Γ $^{µ}$( p, p + q ), S − $^{1}$( p + q ) and ˜ H ( p, p + q ) expansions as IV. SLAVNOV-TAYLOR IDENTITIES AT ONE-LOOP: EXPLICIT CALCULATIONS A. The vectorial vertex function iD − 1 $_{µν}$( q ) = − i { ( q $^{2}$g$_{µν}$ − q$_{µ}$q$_{ν}$ ) [1 + Π$_{T}$( q $^{2}$)] − q$_{µ}$q$_{ν}$ α − Π$_{θ}$( q $^{2}$)˜ q$_{µ}$ ˜ q$_{ν}$ } , (65) q $^{µ}$D − 1 $_{µν}$( q ) = − 1 α q $^{2}$q$_{ν}$ . (66) Γ$_{µνλ}$( p, q, k ) = (2 π ) $^{4}$δ$^{4}$( p + q + k ) ˜ Γ$_{µνλ}$( p, q, − p − q ) (67) G$_{λν}$ ′ ( p, q, k ) = (2 π ) $^{4}$δ$^{4}$( p + q + k ) ˜ G$_{λν}$ ′ ( p, q, − p − q ) , (68) p µ ˜ Γ$_{µνλ}$( p, q, k )[1 + b ( p $^{2}$)] = ˜ G$_{λν}$ ′ ( p, q, k ) { ( q $^{2}$g ν ′ ν − q ν $^{′}$q$_{ν}$ )[1 + Π$_{T}$( q $^{2}$)] + Π$_{θ}$ ( q $^{2}$)˜ q ν $^{′}$˜ q$_{ν}$ } (69) + ˜ G$_{νλ}$ ′ ( p, k, q ) { [ k $^{2}$g λ ′ λ − k λ $^{′}$k$_{λ}$ ][1 + Π$_{T}$( k $^{2}$)] + Π$_{θ}$ ( k $^{2}$) ˜ k λ $^{′}$˜ k$_{λ}$ } . S = ∫ d $^{4}$x [ − Z$_{3}$ 4 G$_{µν}$G µν − ieZ$_{1}$ 2 [ A$_{µ}$, A$_{ν}$ ]$_{⋆}$ G µν + e $^{2}$Z$_{4}$ 4 [ A$_{µ}$, A$_{ν}$ ]$_{⋆}$[ A $^{µ}$, A$^{ν}$]$_{⋆}$ + Z$_{2}$ ¯ ψi/ ∂ψ − ( m + δm ) ¯ ψψ + e Z$_{1}$$_{F}$ ¯ ψ ⋆ / Aψ − 1 2 α ( ∂$_{µ}$A $^{µ}$) 2 + ˜ Z$_{3}$∂$_{µ}$ ¯ C∂ $^{µ}$C − ie ˜ Z$_{1}$∂$_{µ}$ ¯ C ⋆ [ A $^{µ}$, C ]$_{⋆}$ ] , (70) ˜ Γ $^{µ}$( p, p + q ) = Z$_{1}$$_{F}$ γ $^{µ}$eip ∧ q + Λ $^{µ}$( p, p + q ) e ip ∧ $^{q}$, (71) ˜ H$_{i}$ ( p, p + q ) = [ Z$_{i}$$_{+4}$e ip ∧ q + B$_{i}$ ( p, p + q ) e ip ∧ q ] for i = 1 , 2 (72) 11