Using Covariant Polarisation Sums in QCD

TL;DR

This paper investigates the use of covariant polarisation sums in QCD, revealing that common Feynman rules for ghosts are only valid for two external ghosts, and provides a pedagogical derivation of ghost contributions using the optical theorem.

Contribution

It demonstrates that the standard Feynman rule for ghost contributions in QCD is only valid for two ghosts, offering a detailed derivation and correction for cases with more ghosts.

Findings

Feynman rule for ghosts is only valid for two external ghosts.

Ghost cross terms cannot be simplified into squared amplitudes for more than two ghosts.

Provides a pedagogical derivation of ghost contributions using the optical theorem.

Abstract

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.