For an illustration [189, 191, 190] consider a truncated solution of the FRGE (C.11),

To begin with we neglect the running of the kinetic term (“local potential ansatz”) and set . For functionals of this type, and in a momentum basis where corresponds to , the denominator appearing under the trace of Equation (C.11) reads Here we used a simple mass-type cutoff which is sufficient to make the point. In a loop calculation of it is the inverse of Equation (C.23), evaluated at the vacuum expectation value which appears as the effective propagator in all loops. It contains an IR cutoff at the scale , a mass term which adds to in the special case considered here. (In general introduces a -dependent mass.)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 ,

Equation (C.24) is an extremely useful tool for effectively going beyond the truncation (C.22) without having to derive and solve a more complicated flow equation. In fact, thanks to the additional -dependence which comes into play via , Equation (C.24) can predict certain terms which are contained in even though they are not present in the truncation ansatz.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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