Taking the -derivative of Equation (C.6) with Equations (C.1, 2.10) inserted one finds
We add some comments on the FRGE (C.11):
To see this let us momentarily write for the kernel of in Fourier space. By assumption is a family of functions which remains pointwise bounded as (but the falloff in may not be strong enough so as to define a bounded operator in the limit). The right-hand-side of Equation (C.11) then is proportional to , which by Equation (C.3) behaves like for . On the other hand by Equation (C.4) the derivative has support mostly on a thin shell around , so that the (potentially problematic) large behavior of is irrelevant.
However, there are conceptual differences between the effective average action and a genuine Wilsonian action , as discussed in Appendix A.
First, in the literature the running scale on which a Wilsonian action depends is frequently referred to as an ultraviolet cutoff and is denoted by . This is due to a difference in perspective: If all modes of the original system with momenta between infinity (or ) are integrated out, is an infrared cutoff for the original model, but it plays the role of an UV cutoff for the “residual theory” of the modes below this scale, which are to be integrated out still. For them has the status of a bare action.
Second, the Wilsonian action describes a set of different actions, parameterized by and subject to a flow equation like Equation (A.4), for one and the same system; the Greens functions are independent of and have to be computed from by further functional integration. In contrast the effective average action can be thought of as the standard effective action for a family of different systems; for any value of it equals the standard effective action (generating functional for the vertex- or 1-PI Green’s functions) for a model with bare action . The latter is of course not subject to a Wilsonian type flow equation like Equation (A.4). In particular the multi-point functions do depend on . This is a desired property, however, as these -dependent Green’s functions are supposed to provide an effective field theory description of the physics at scale , without further functional integration. See  for a detailed discussion.
There exists a variety of different functional renormalization group equations. We refer to [21, 166, 29, 146, 229] for reviews. To a certain extent they contain the same information but encoded in different ways (see e.g. ); the differences become important in approximations (the ‘truncations’ described below) where simple truncations adapted to a certain application in one FRGE might correspond to more complicated and less adapted truncations in another. We use the effective average action [228, 229, 29] here because of its effective field theory properties and because via the background field method it has been extended to gravity (see ). FRGEs invariant under field reparameterizations have been developed in [165, 44] but have not yet been applied in computations.
Concretely the truncation is usually done by assuming an ansatz of the form
Another approximation procedure for the solution of Equation (2.12) is the local potential approximation [21, 111, 147]. Here the functionals ’s are constrained to contain only a standard kinetic term plus arbitrary non-derivative terms. Generally convexity can be used as guideline to identify good truncations.
A slight extension of the local potential approximation is to allow for a (-independent) wave function renormalization, i.e. a running prefactor of the kinetic term: . Using truncations of this type one should employ a slightly different normalization of , namely for . Then combines with to the inverse propagator , as required if the IR cutoff is to give rise to a of size rather than . In particular in theories with more than one field it is important that all fields are cut off at the same . This is achieved by a cutoff function of the form
© Max Planck Society and the author(s)