The Ward Identity from the Background Field Dependence of the Effective Action

The dependence of the effective action for gauge theories on the background field obeys an exact identity. We argue that for Abelian theories the Ward identity follows from the more general background field identity. This observation is particularly relevant for the anomalous Ward identity valid for gauge theories with an effective infrared cutoff as used for flow equations. E-mail: [email protected] E-mail: [email protected] The effective average action Γk is a useful concept for the investigation of infrared problems. In field theory it corresponds to the quantum effective action with an effective infrared cutoff ∼ k for the fluctuations. In statistical physics it is the coarse grained free energy with coarse graining length scale ∼k. Exact non-perturbative flow equations describe the dependence of Γk on k [1]. They are related to the Wilsonian approach to the renormalization group equations [2]. Since non-Abelian gauge theories are plagued in perturbation theory by severe infrared problems the use of the effective average action seems particularly promising here. Writing down exact flow equations for gauge theories poses no additional difficulty. The problem of correct implementation of gauge symmetry arises rather on the level of their solutions. In fact, models with local gauge symmetry correspond to particular trajectories in the space of general solutions of the flow equations. It is crucial to find appropriate identities for Γk which enforce a restriction to “gaugeinvariant solutions” and guarantee gauge invariance for k → 0. So far, two different lines of research have been followed in this respect. The first one [3][4] works in the background field formalism where Γk[A, Ā] depends on the classical gauge field Aμ (conjugate to the source) and the background gauge field Āμ which enters through the gauge fixing and the infrared cutoff. By construction Γk[A, Ā] is invariant under simultaneous gauge transformations of Aμ and Āμ. This combined gauge invariance is, however, not sufficient to guarantee that the solution lies on a trajectory appropriate for a gauge-invariant theory. An additional exact identity for the background field dependence δΓk/δĀμ was derived [4] and it is believed that this identity guarantees full gauge invariance of the theory. The second approach [5][6] centers on the Ward-Takahashi or Slavnov-Taylor identities. These identities receive anomalous contributions [6] from the presence of the infrared cutoff term ∼ Rk which vanish only for k → 0. Obviously, the same type of identities can also be derived within the background field formalism. One would suspect that the background field identity and the Ward identity are not unrelated since both reflect the content of gauge symmetry. Their exact relation has, however, not been revealed in the past. 1 In this letter we indicate how the Ward identity can be derived from the background field dependence of the effective action for Abelian gauge theories. The model we use is scalar electrodynamics (SQED) in arbitrary dimension d. The classical action consists of the usual SQED action plus a gauge-fixing term and a quadratic term implementing an infrared cutoff. For the complex scalar field χ(x) the infrared cutoff reads [4] ∆ (S) k S = ∫ dx χ∗(x)Rk(D[Ā])χ(x). (1) Here Rk is a function that cuts off the modes with momentum smaller than k with Rk(0) ∼ k . More precisely, the cutoff distinguishes between eigenvalues of the covariant Laplacian, D[Ā] = −(∂μ + igĀμ(x))(∂ μ + igĀ(x)), where Āμ(x) is the background gauge field. The complete set of quadratic cutoff terms includes a term for the gauge sector which can be chosen to be gauge invariant in Abelian gauge theories. If this is the case it does not affect the Ward identity or the background field identity and does not need to be specified for our discussion. We choose the background gauge fixing Γgf [A, Ā] = 1 2α ∫