Gauge invariance and renormalization

Abstract

The renormalization of Abelian and non-Abelian local gauge theories is discussed. It is recalled that whereas Abelian gauge theories are invariant to local c-number gauge transformations δA μ ( x) = ∂ μ ,…, with □ Λ = 0, and to the operator gauge transformation δA μ ( x) = ∂ μ φ( x), …, δφ( x) = α −1 ∂· A( x), with □ φ = 0, non-Abelian gauge theories are invariant only to the operator gauge transformations δA μ(x) ∼ ▪ μC(x) , …, introduced by Becchi, Rouet and Stora, where ▪ μ is the covariant derivative matrix and C is the vector of ghost fields. The renormalization of these gauge transformation is discussed in a formal way, assuming that a gauge-invariant regularization is present. The naive renormalized local non-Abelian c-number gauge transformation δA μ(x) = (Z 1/Z 3)gA μ(x) × Λ(x)+∂ μ Λ(x) , …, is never a symmetry transformation and is never finite in perturbation theory. Only for Λ(x) = (Z 3/Z 1)L with L finite constants or for Λ(x) = Ω z ̃ 3C(x) with Ω a finite constant does it become a finite symmetry transformation, where z ̃ 3 is the ghost field renormalization constant. The renormalized non-Abelian Ward-Takahashi (Slavnov-Taylor) identities are consequences of the invariance of the renormalized gauge theory to this formation. It is also shown how the symmetry generators are renormalized, how photons appear as Goldstone bosons, how the (non-multiplicatively renormalizable) composite operator A μ × C is renormalized, and how an Abelian c-number gauge symmetry may be reinstated in the exact solution of many asymptotically fr ee non-Abelian gauge theories.