Color average (source code)

= Color average
{title2=$\overline{|\mathcal M|^2}_{\rm color}=N^{-2}\sum_{i,j,k,l}|\mathcal M_{ij;kl}|^2$}

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>, $c_s=T^a_{ji}T^a_{kl}$ gives $N^{-2}\sum|c_s|^2=(N^2-1)/(4N^2)$. For the exchange contraction $c_t=T^a_{ki}T^a_{jl}$, $N^{-2}\sum c_sc_t^*=-(N^2-1)/(4N^3)$ by the <SU(N) fundamental completeness relation>. The negative overlap reverses the sign of the corresponding Abelian <interference> term.