Flow Equations and BRS Invariance for Yang-Mills Theories

Flow equations describe the evolution of the effective action Γk in the process of varying an infrared cutoff k. The presence of the infrared cutoff explicitly breaks gauge and hence BRS invariance. We derive modified Slavnov-Taylor identities, which are valid for nonvanishing k. They guarantee the BRS invariance of Γk for k → 0, and hence allow the study of non-abelian gauge theories by integrating the flow equations. Within a perturbative expansion of Γk, we derive an equation for a k dependent mass term for the gauge fields implied by the modified Slavnov-Taylor identities. Supported by a DFG Heisenberg fellowship; e-mail: I96 at VM.URZ.UNI-HEIDELBERG.DE The use of flow equations (or exact renormalization group equations [1] resp. evolution equations) in continuum quantum field theory has recently been the subject of numerous investigations. They were employed by Polchinski and many others [2]-[6] in order to simplify proofs of perturbative renormalizability. Here the flow equations are used to construct bare actions (depending on an ultraviolet cutoff Λ), which guarantee the existence of finite quantum effective actions in the limit Λ → ∞. Alternatively the flow equations can be used to construct quantum effective actions in terms of bare actions, with an ultraviolet cutoff Λ kept fixed. Their integration with respect to an infrared cutoff k then serves as a computational tool, which allows to calculate the generating functionals (or the Green functions) in the limit k → 0 in terms of boundary conditions for those functionals at some large infrared cutoff k = k̄. The result corresponds to the one of a quantum field theory with fixed ultraviolet cutoff Λ = k̄, with a classical action related to the boundary conditions at k = k̄. For recent work in this direction see refs. [7],[8]. The simple form and the exactness of the flow equations is based on the fact that the cutoffs are introduced by modifying the propagators of the fields. In momentum space, e.g., the propagators get multiplied by a function of p, which vanishes rapidly for momenta beyond the cutoffs. The application of the present concept of flow equations to gauge theories leads thus to serious problems, since the presence of such cutoffs breaks gauge invariance explicitly. (Kleppe and Woodard [9] studied a closely related regularization which affects not just the propagators, but modifies the entire action. This method preserves distorted versions of gauge symmetries, but the resulting effective action no longer satisfies simple flow equations.) In [8] background gauge fields were introduced in order to cope with this situation, but their presence in the final expressions for the generating functionals leads to new conceptual and practical difficulties. In [4]-[6] Ward or Slavnov-Taylor (ST) identities [10] were employed in order to obtain proofs of perturbative renormalizability for gauge theories. Let us denote by Gk (J), Γ Λ k (φ) the generating functionals of connected and oneparticle irreducible Green functions, respectively. The indices k and Λ refer to the fact that only modes with k < p < Λ have been integrated out, i.e. Gk and Γ Λ k are computed in the presence of an infrared cutoff k and an ultraviolet cutoff Λ in the propagator. For k = Λ GΛ and Γ Λ Λ are simply related to a bare action S0(Λ) [3], [7]. Proofs of perturbative renormalizability start with the construction of GΛ resp. ΓΛ and hence S0(Λ), given the knowledge of the relevant or marginal couplings of G0 and Γ Λ 0 , by integrating the flow equations for G Λ k resp. Γ Λ k . Subsequently the limit Λ → ∞ has to be shown to exist, keeping G0 and Γ Λ 0 finite. In [4]-[6] Ward or Slavnov-Taylor (ST) identities were imposed on the relevant or marginal couplings of G0 resp. Γ Λ 0 , and it was argued, that a bare action S0(Λ) consistent with these identities can be constructed. This procedure is of no help, however, if we want to compute the generating functionals for k → 0 in terms of S0(Λ); we do not know the relevant or marginal couplings of G0 resp. Γ Λ 0 beforehand. It should be clear that G Λ Λ resp. Γ Λ Λ and hence S0(Λ) cannot satisfy the standard ST identies, which are related to BRS