Background field quantization in non-covariant gauges: renormalization and WTST identities

Abstract

Background field quantization of pure YM theories in non-covariant gauges is treated with particular emphasis on renormalization. Gauge fixing terms of the form ( 1 2α )n · Q aƒ abn · Q b are considered where ƒ ab can assume the forms ƒ ( i) ab = −δ ab (the axial gauge), ƒ ( ii) ab = (n · D(A)) 2ab/n 4 and ƒ ( iii) ab = D 2(A) ab/n 2 (the planar gauge). For the cases where ƒ ab depends explicitly on the background field A μ a the ghost sector is enlarged by the addition of appropriate Nielsen-Kallosh ghost fields. The BRS identities for these gauge choices are derived and solved. The quantum-corrected versions of both the bare background field gauge transformations and the bare quantum field gauge transformations are obtained from the BRS analysis. It is also shown that, to one loop, all the counter terms are determined by the background field independent part of the theory and this result is used, in cases (ii) and (iii), to derive all the counter terms and to show that Kallosh's theorem is verified. The result is also used to demonstrate the pathological nature of case (i) for α ≠ 0, in particular the result that Kallosh's theorem is not applicable. The result that the generating functional of Green functions is independent of the background field A μ a in the absence of all external sources is generalized to the case of non-covariant gauges. The equality established by Abbott between the 1PI generating functionals Γ[A, 0] and Γ c [ Q; A] Q = A , where Γ c is a conventional generating functional in an A-dependent gauge, is analysed. We show that the WTST identities satisfied by Γ c reduce, when Q is set equal to A, to the naive Ward-identity satisfied by Γ[A, 0] .