Abstract
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 }, 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 )^* between ghost amplitudes cannot be transformed into (-1)^{n/2}|mathcal {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.
Highlights
The traditional way to derive physical observables like decay widths and cross sections from Feynman amplitudes A is to calculate the squared amplitude |A|2 and to sum over the final and to average over the initial spin and polarisation states of the external particles
We examine the use of covariant gauges for external gauge bosons and in particular the Feynman rules for external ghosts
We have reviewed the use of covariant gauges for the polarisation states of external gauge bosons
Summary
The traditional way to derive physical observables like decay widths and cross sections from Feynman amplitudes A is to calculate the squared amplitude |A|2 and to sum over the final and to average over the initial spin and polarisation states of the external particles. For particles with spin s ≥ 1, the summation over polarisation states is complicated by the fact there is a mismatch between the physical number of polarisation states and field variables. A massless spin-one field Aμ has four components but only two physical polarisation states. Excluding the non-physical states requires to choose a non-covariant form of the polarisation sum, what in turn results in lengthy intermediate expressions for the squared amplitude |A|2. The squared amplitude of the four-gluon amplitude in QCD contains 228,420 terms before simplifications reduce them to four
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