Some general properties of the effective average action have been anticipated in Section 2.2 in relation to the UV renormalization problem. In this appendix, we discuss the effective average action and its properties in more detail. First we give the definition of from the functional integral with suitable mode cutoff kernels . Then we derive the basic flow equation (2.11) and discuss its properties. Finally the decoupling properties of are exemplified, which are useful in particular for “renormalization group improvement”.
Again we illustrate the concepts below for a scalar quantum field theory on flat space. Part of the rationale for using the effective average action, however, stems from the fact that via the background field method it can be generalized to gauge theories as well. See  for an alternative computationally tested approach.
For orientation we briefly describe the route that led to the effective average action and its generalization to gauge theories. Initially the average action proper [193, 186, 185] was introduced as the generating functional for the correlators of fields averaged over a Euclidean spacetime volume , in dimensions. Here “averaging” is to be understood in the literal sense; the defining functional integral (over a scalar , say) contains a smeared delta functional which forces , the average of over a ball of radius , to be equal to an externally prescribed field . This construction is a continuum counterpart of a Kadanoff block spin transformation. For the average action proper approaches the constraint effective potential studied earlier .
While the average action proper for non-gauge theories has a clear physical interpretation it has proven difficult to generalize it to gauge theories. With certain modifications this is possible in the Abelian case [186, 185] but the construction fails for non-Abelian Yang–Mills theories. This was the motivation for introducing the effective average action [228, 186, 185, 187]. It reinterprets the averaging in the non-gauge case as a cutting off of Fourier modes, the eigenfunctions of the ordinary Laplacian, and replaces it in the Yang–Mills case by a corresponding cuting-off of the eigenmodes of the covariant Laplacian . Contrary to the old average action the new one is defined in terms of a Legendre transform and therefore encodes the information about the multi-point functions in the more condensed 1PI form (see also ). The price to pay is that the simple averaging is replaced with the less intuitive weighing of field modes according to their eigenvalue, with the corresponding change in the meaning of “long” and “short”wavelength modes. However the effective average action has better effective field theory properties and satisfies a closed functional evolution equation. Both aspects have been tested in Yang–Mills theories, we refer to [229, 29] for a review and further literature.
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