Calculation of the vacuum polarization in non-covariant gauges incorporating Nielsen-Kallosh ghosts

Abstract

The background field quantization procedure for pure YM theories is used in conjunction with non-covariant gauges characterized by the gauge fixing term ( 1 2 α)n·Q af i abn·Q b where f i ab can asume the forms f I = δ ab (i.e. axial gauge), f II ab = ( n·D( A)) 2 ab /( n 2) 2 or f III = D 2( A ab / n 2 (i.e. planar gauge) where n 2 ≠ 0. Here A μ a and Q μ a represent respectively the classical background field and the quantum field. It is noted that if f i ab explicitly depends on the background field, then it is necessary to introduce Nielsen-Kallosh ghosts in addition to the expected Faddeev-Popov ghosts. Explicit calculations to one-loop order show that for f II and f III, the divergent part of the vacuum polarization is [( i/16π 2)C 2δ ab g 2/(2−ω)] 11 3 (p 2δ μν−p μp ν) , while in the axial gauge the vacuum polarization is transverse but α- and n-dependent. The latter result - an apparent contradiction of Kallosh's theorem — is shown to arise due to the unconventional asymptotic behaviour of the vector propagator in the axial gauge.