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A uniform incoherent initial mixture of two independently prepared fundamental color charges requires an average over their $N^2$ initial possibilities, together with a sum over unobserved final color charges. This differs from preparing a coherent color singlet. For quark-antiquark annihilation through one vector in the adjoint representation, $c_s=T_{ji}^a{T}_{kl}^a$ gives $N^{-2}\sum |c_s{|}^2=(N^2-1)/(4N^2)$. For the exchange contraction $c_t=T_{ki}^a{T}_{jl}^a$, $N^{-2}\sum c_s{c}_t^{\ast }=-(N^2-1)/(4N^3)$ by the SU(N) fundamental completeness relation. The negative overlap reverses the sign of the corresponding Abelian interference term.

Both $s$-channel annihilation and $t$-channel gluon exchange are now allowed. Their spin sum factors are the Bhabha scattering factors with $e\mapsto g$. Their color charge tensors, with the external indices of Solution, areEach squared term has $K_{ss}=K_{tt}=(N_c^2-1)/(4N_c^2)$. The cross term instead has the color averageTo evaluate it, use the SU(N) fundamental completeness relationThis follows by completing the orthogonal basis of traceless Hermitian matrices with the identity. Since $\mathrm{tr}{T}^b=0$, it gives ${\sum }_a{T}^a{T}^b{T}^a=-T^b/(2N_c)$, and henceMultiplying the two squared Bhabha scattering terms by $K_{ss}$ and its cross term by $K_{st}$ givesFor the SU(3) gauge group, this isThe negative color charge overlap reverses the sign of the QED interference. Since $st<0$, this cross contribution is positive in the physical region. No additional spinor calculation is needed; all the new information is in the SU(N) fundamental completeness relation.