### C.2 Flow equation

As anticipated, the scale dependence of is governed by a functional differential equation [22822929],
This is the functional renormalization group equation (FRGE) for the effective average action. The ingredients have already been explained in Section 2.3; here we present its derivation. An instructive derivation of Equation (C.11) starting directly from Equation (C.8) can be found in [44] (see also [152]). A technically quicker one proceeds as follows [228]:

Taking the -derivative of Equation (C.6) with Equations (C.1, 2.10) inserted one finds

with the and -dependent expectation values defined as in Equation (C.2). Next it is convenient to introduce the connected 2-point function and . Since and are related by a Legendre transformation, and are mutually inverse integral kernels, i.e.  for the associated integral operators. Taking two -derivatives of Equation (C.1) one obtains . Substituting this expression for the two-point function into Equation (C.12) we arrive at
In terms of this translates into . The derivation is completed by noting that , where the second equality follows by differentiating Equation (C.7).

1. The right-hand-side of Equation (C.11) can be also rewritten in the form of a one-loop expression
Here the scale derivative acts only on the -dependence of , not on . The expression in Equation (C.11) differs from a standard one-loop determinant in two ways: It contains the Hessian of the actual effective action rather than that of the bare action , and it has a built in IR regulator . These modifications make Equation (C.11) an exact equation.
2. Observe that the FRGE (C.6) is independent of the bare action , which enters only via the initial condition (for large ). In the FRGE approach the calculation of the path integral for is replaced with the task of integrating this RG equation from , where the initial condition is imposed, down to , where the effective average action equals the ordinary effective action . The role of the bare action in the removal of cutoff and its relation to the UV renormalization problem has already been decribed in Section 2.3. Here we repeat again that the explicit dependence entering via is harmless, and can essentially be replaced with .

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.

3. By repeated differentiation of Equation (C.11) and evaluation at an extremizing configuration one obtains a coupled infinite system of flow equations for the -point functions (B.7). For example for a translation invariant theory [34],
Here we defined in analogy to Equation (B.7) and wrote for its Fourier transform. Similarly for . One sees that lower multipoint functions couple to higher order ones in a way so that only the infinite system closes. However if all external momenta are small, , the insertion will ensure that also the internal momentum is small. This is the rationale for the derivative (small momentum) expansion. For an approximation suited for uniformly large momenta see [34].
4. The above FRGE is in the spirit of Wilson–Kadanoff renormalization ideas, but with the iterated coarse graining procedure replaced by a direct mode cutoff. Since the kernel in Equation (A.2) cuts off momenta with , the right-hand-side of Equation (A.2) corresponds to Equation (C.2) evaluated at . One could also have derived a flow equation for (a suitable variant of) .

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 [29] for a detailed discussion.

There exists a variety of different functional renormalization group equations. We refer to [2116629146229] for reviews. To a certain extent they contain the same information but encoded in different ways (see e.g. [147]); 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 [22822929] here because of its effective field theory properties and because via the background field method it has been extended to gravity (see [179]). FRGEs invariant under field reparameterizations have been developed in [16544] but have not yet been applied in computations.

5. To solve the functional flow equation (C.11) approximations are indispensible. One which does not rely on a perturbative expansion is by truncation of the space of candidate continuum functionals to one where the initial value problem for the flow equation (2.12) can be solved in reasonably closed form. In this case one can then by ‘direct inspection’ determine the initial data for which a global solution exists. The existence of a nontrivial unstable manifold for can then be taken as witnessing the renormalizability of an implicitly defined ‘hierarchical’ dynamics (see Section 2.3).

Concretely the truncation is usually done by assuming an ansatz of the form

where the ’s are scale dependent (‘running’) parameters as in the previous general discussion, and the -independent functionals span the subspace selected. The ‘art’ of course consists in choosing a set of small enough to be computationally manageable and yet such that the projected flow encapsulates the essential physics features of the exact flow. The projected RG flow then is described by a set of ordinary differential equations for the parameters . Schematically those equations arise as follows. Let us assume the finite set , , can be extended to a ‘basis’ (in the sense discussed in Section 2.1) , , of the full space of functionals. Expanding the dependence of on the right-hand-side of the FRGE (C.11) in this basis an expression of the form
arises. Here the play the role of generalized beta functions for the parameters , . Neglecting, (i) the terms with in Equation (C.17) and (ii) the dependence of on the parameters , , a closed system of ordinary differential equations arises: , . As in Equation (A.3) one can reparameterize the in Equation (C.14) in terms of inessential parameters and couplings , , with the latter ones made dimensionless. The corresponding functions then become the beta functions proper and the system of differential equations reads
The ’s have no explicit dependence and define a ‘time independent’ vector field on the space of couplings .

Another approximation procedure for the solution of Equation (2.12) is the local potential approximation [21111147]. Here the functionals ’s are constrained to contain only a standard kinetic term plus arbitrary non-derivative terms

Since the potential function could be (Taylor-) expanded one can view (C.19) as a simple infinite parametric version of Equation (C.16). The truncated flow equations for Equation (C.19) now amount to a partial differential equation (in two variables) for the running potential . It is obtained by inserting Equation (C.19) into the FRGE and projecting the trace onto functionals of the form (C.19). This is most easily done by inserting a constant field into both sides of the equation since it gives a nonvanishing value precisely to the non-derivative ’s. As , has no explicit dependence the trace is easily evaluated in momentum space. This leads to the following partial differential equation:
This equation describes how the classical (or microscopic) potential evolves into the standard effective potential . Remarkably, for an appropriate choice of the limit is automatically a convex function of , a feature the full effective action must have but which is usually destroyed in perturbation theory. For a detailed discussion of this point we refer to [29]. 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

with as in Equation (4.15). Here is in general a matrix in field space. In the sector of modes with inverse propagator the matrix is chosen diagonal with entries . In a scalar field these factors are automatically positive and the flow equations in the various truncations are well-defined.