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 [29
] 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. [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 [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 [179]). 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
where theAnother 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
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
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