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, $$\sum _{\lambda } \varepsilon _\mu ^{(\lambda )}\varepsilon _\nu ^{(\lambda )*} = -\eta _{\mu \nu }$$ ∑ λ ε μ ( λ ) ε ν ( λ ) ∗ = - η μ ν , 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 $$\mathcal {A}(c_1,\bar{c}_1;\ldots )\mathcal {A}(\bar{c}_1,c_1;\ldots )^*$$ A ( c 1 , c ¯ 1 ; … ) A ( c ¯ 1 , c 1 ; … ) ∗ between ghost amplitudes cannot be transformed into $$(-1)^{n/2}|\mathcal {A}(c_1,\bar{c}_1;\ldots )|^2$$ ( - 1 ) n / 2 | A ( c 1 , c ¯ 1 ; … ) | 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.