Extended BRS Symmetry and Gauge Independence in On-Shell Renormalization Schemes

Extended BRS symmetry is used to prove gauge independence of the fermion renormalization constant Z2 in on-shell QED renormalization schemes. A necessary condition for gauge independence of Z2 in on-shell QCD renormalization schemes is formulated. Satisfying this necessary condition appears to be problematic at the three-loop level in QCD. In on-shell schemes, the fermion mass renormalization Zm and wave function renormalization Z2 have been observed to be gauge parameter independent in explicit two-loop QED and QCD calculations [1]. Gauge parameter independence of Z2 is phenomenologically significant because it implies that the difference between the (fermion) anomalous dimension of heavy quark effective theories [2] and QCD is gauge independent. An extension of BRS symmetry, which allows variations of the gauge parameter to be included as part of the symmetry transformations [3], will be applied to the gauge parameter dependence of Z2. This approach results in an extension of Slavnov-Taylor identities, allowing gauge dependence to be formulated algebraically. Previous application of these techniques resulted in a proof of the gauge independence of the mass renormalization Zm to all orders in on-shell QED and QCD renormalization schemes [4]. We will prove the gauge parameter independence of Z2 in on-shell schemes for QED and formulate a necessary condition for gauge independence in QCD which appears problematic beyond the two-loop level. This complements earlier work on gauge independence of Z2 in QED resulting in the (dimensionally-regularized) relation [5] ∂Z2 ∂ξ ∼ ∫ dk 1 k4 = 0 (1) where ξ is the gauge parameter and the massless tadpole is zero in dimensional regularization. Since the QED result (1) cannot be extended to QCD, our extended BRS symmetry proof for QED provides a new approach to formulating questions of gauge independence of Z2 in QCD. The QED Lagrangian in the auxiliary field formalism [6] for covariant gauges is L = − 1 4 F 2 + ψ̄ (iD/−m)ψ + ξ 2 B +B∂ ·A− c̄∂c (2) where F is the field strength and B is the auxiliary gauge field. This Lagrangian is invariant under the BRS symmetry δAμ = ǫ∂μc , δψ̄ = iǫgcψ̄ , δc = 0 δB = 0 , δψ = −iǫgcψ , δc̄ = 0 (3) email: [email protected]