as follows. First, the free propagators are the same as in the commutative version of the model, i.e., for the fermion, photon and ghost field propagators, respectively. Introducing the notation p ∧ k ≡ 1 $_{2}$ξθ $^{µν}$p$_{µ}$ k$_{ν}$ , we determine the vertices as being where γ $^{µνα}$( p, q, k ) = ( p − q ) $^{α}$gµν + ( q − k ) $^{µ}$gνα + ( k − p ) $^{ν}$g$^{αµ}$. 4 p = i / p − m , (6) p GLYPH<c109> GLYPH<c110> = − i p 2 [ g µν − (1 − α ) p$_{µ}$p$_{ν}$ p 2 ] , and (7) p = i p $_{2}$, (8) p q k GLYPH<c109> = − ieγ $^{µ}$eip ∧ $^{k}$, (9) q p k GLYPH<c108> GLYPH<c110> GLYPH<c109> = 2 e sin( p ∧ q ) γ $^{µνα}$( p, q, k ) , (10) = − 4 ie 2 [ ( g $^{µβ}$gαν − g $^{µν}$g$^{αβ}$) sin( p ∧ r ) sin( q ∧ k ) +( g $^{µν}$gαβ − g $^{µα}$g$^{νβ}$) sin( q ∧ p ) sin( r ∧ k ) (11) + ( g $^{µα}$gνβ − g $^{µβ}$g$^{αν}$) sin( p ∧ k ) sin( r ∧ q ) ] , p q k GLYPH<c109> = 2 ek $^{µ}$sin( p ∧ k ) , (12)