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 .
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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