Using Covariant Polarisation Sums in QCD

Covariant gauges lead to spurious, non-physical polarisation states of gauge bosons. In QED, the use of the Feynman gauge, $\sum_{\lambda} \epsilon_\mu^{(\lambda)}\epsilon_\nu^{(\lambda)\ast} = -\eta_{\mu\nu}$, is justified by the Ward identity which ensures that the contributions of non-physical polarisation states cancel in physical observables. In contrast, the same replacement can be applied only to a single external gauge boson in squared amplitudes of non-abelian gauge theories like QCD. In general, the use of this replacement requires to include external Faddeev-Popov ghosts. We present a pedagogical derivation of these ghost contributions applying the optical theorem and the Cutkosky cutting rules. We find that the resulting cross terms $A(c_1,\bar{c}_1;\ldots)A(\bar{c}_1,c_1;\ldots)^\ast$ between ghost amplitudes cannot be transformed into $(-1)^{n/2}|A(c_1,\bar{c}_1;\ldots)|^2$ in the case of more than two ghosts. Thus the Feynman rule stated in the literature holds only for two external ghosts, while it is in general incorrect.