For an illustration [189, 191
, 190
] consider a truncated solution of the FRGE (C.11
),
The -modes (plane waves) are integrated out efficiently only in the domain
.
In the opposite case all loop contributions are suppressed by the effective mass square
. It is the sum of the “artificial” cutoff
, introduced in order to effect the
coarse graining, and the “physical” cutoff terms
. As a consequence,
displays a significant dependence on
only if
because
otherwise
is negligible relative to
in all propagators; it is then the physical
cutoff scale
which delimits the range of
-values which are integrated
out.
Typically, for very large,
is larger than the physical cutoffs so that
“runs” very fast.
Lowering
it might happen that, at some
, the “artificial” cutoff
becomes smaller than the
running mass
. At this point the physical mass starts playing the role of the actual cutoff; its effect
overrides that of
so that
becomes approximately independent of
for
. As a result,
for all
below the threshold
, and in particular the ordinary effective action
does not differ from
significantly. This is the prototype of a “decoupling” or “freezing”
phenomenon [208].
The situation is more interesting when is negligible and
competes with
for
the role of the actual cutoff. (Here we assume that
is
-independent.) The running of
, evaluated at a fixed
, stops once
where the by now field dependent
decoupling scale obtains from the implicit equation
. Decoupling occurs for
sufficiently large values of
, the RG evolution below
is negligible then; hence, at
,
A simple example illustrates this point. For large, the truncation (C.22
) yields a logarithmic
running of the
-coupling:
. As a result, Equation (C.24
) suggests that
should
contain a term
. Since, in leading order,
, this leads us to the prediction of a
-term in the conventional effective action. This prediction, including the prefactor of the term, is
known to be correct: The Coleman–Weinberg potential of massless
-theory does indeed contain this
-term. Note that this term is not analytic in
, so it lies outside the space of functionals
spanned by the a power series ansatz like Equation (C.22
).
This example illustrates the power of decoupling arguments. They can be applied even when is
taken
-dependent as it is necessary for computing
-point functions by differentiating
. The
running inverse propagator is given by
, for example. Here a new
potential cutoff scale enters the game: the momentum
dual to the distance
. When it
serves as the operative IR cutoff in the denominator of the multiply differentiated FRGE, the
running of
, the Fourier transform of
, stops once
is smaller than
. Hence
for
, provided no other physical scales
intervene. As a result, if one allows for a running
-factor in the truncation (C.22
) one predicts a
propagator of the type
in the standard effective action. Note that generically it
corresponds to a nonlocal term
in
, even though the truncation ansatz was
local.
In the context of the effective average action formalism for gravity this kind of reasoning [135, 134] also
underlies the evaluation of the UV behavior of the propagators in the “anomalous dimension argument” of
Section 2.4. If is approximately constant, the graviton
-factor varies as
, and the
corresponding propagator
is proportional to
in momentum space and to
in position space.
In the literature similar arguments have been used for the “renormalization group improvement” of cosmological [37, 38, 183, 28] and black hole spacetimes [36, 35, 40] on the basis of the effective average action (see also [189, 191, 190] for a discussion of different improvement schemes).
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