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C.3 Decoupling properties

As compared to other generating functionals the effective average action has particularly benign decoupling properties. These are crucial for practical applications and provide a powerful tool for extracting physics information from Γ Λ,k, in particular in the context of “renormalization group improvement”. In the following we suppress the UV cutoff Λ as it plays no role in the discussion.

For an illustration [189Jump To The Next Citation Point191Jump To The Next Citation Point190Jump To The Next Citation Point] consider a truncated solution of the FRGE (C.11View Equation),

∫ { } Γ [φ] = d4x 1-Z(k) ∂ φ ∂μφ + 1-m2 (k )φ2 + -1-λ(k)φ4 + ... . (C.22 ) k 2 μ 2 12
To begin with we neglect the running of the kinetic term (“local potential ansatz”) and set Z(k) ≡ 1. For functionals of this type, and in a momentum basis where − ∂2 corresponds to p2, the denominator appearing under the trace of Equation (C.11View Equation) reads
Γ (2)+ ℛk = p2 + m2 (k) + k2 + λ(k)φ2 + .... (C.23 ) k
Here we used a simple mass-type cutoff ℛk = k2 which is sufficient to make the point. In a loop calculation of Γ k it is the inverse of Equation (C.23View Equation), evaluated at the vacuum expectation value φ which appears as the effective propagator in all loops. It contains an IR cutoff at the scale k, a mass term 2 k which adds to 2 m (k ) in the special case considered here. (In general 2 ℛk ≡ ℛk(p ) introduces a p2-dependent mass.)

The p μ-modes (plane waves) are integrated out efficiently only in the domain p2 ≳ m2 + k2 + λ φ2 + .... In the opposite case all loop contributions are suppressed by the effective mass square m2 + k2 + λφ2 + .... It is the sum of the “artificial” cutoff k2, introduced in order to effect the coarse graining, and the “physical” cutoff terms m2 (k) + λ(k) φ2 + .... As a consequence, Γ k displays a significant dependence on k only if k2 ≳ m2(k ) + λ (k )φ2 + ... because otherwise 2 k is negligible relative to 2 2 m + λ φ + ... in all propagators; it is then the physical cutoff scale 2 2 m + λφ + ... which delimits the range of 2 p-values which are integrated out.

Typically, for k very large, k2 is larger than the physical cutoffs so that Γ k “runs” very fast. Lowering k it might happen that, at some k = k dec, the “artificial” cutoff k becomes smaller than the running mass m (k). At this point the physical mass starts playing the role of the actual cutoff; its effect overrides that of k so that Γ k becomes approximately independent of k for k < kdec. As a result, Γ k ≈ Γ kdec for all k below the threshold kdec, and in particular the ordinary effective action Γ = Γ 0 does not differ from Γ kdec significantly. This is the prototype of a “decoupling” or “freezing” phenomenon [208].

The situation is more interesting when 2 m is negligible and 2 k competes with 2 λ φ for the role of the actual cutoff. (Here we assume that φ is x-independent.) The running of Γ k, evaluated at a fixed φ, stops once k ≲ kdec(φ ) where the by now field dependent decoupling scale obtains from the implicit equation k2 = λ(kdec)φ2 dec. Decoupling occurs for sufficiently large values of φ, the RG evolution below k dec is negligible then; hence, at k = 0,

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

A simple example illustrates this point. For k large, the truncation (C.22View Equation) yields a logarithmic running of the Φ4-coupling: λ(k) ∝ ln(k). As a result, Equation (C.24View Equation) suggests that Γ should contain a term 4 ∝ ln (kdec(φ))φ. Since, in leading order, kdec ∝ φ, this leads us to the prediction of a φ4 ln(φ)-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 φ4-theory does indeed contain this 4 φ ln(φ)-term. Note that this term is not analytic in φ, so it lies outside the space of functionals spanned by the a power series ansatz like Equation (C.22View Equation).

This example illustrates the power of decoupling arguments. They can be applied even when φ is taken x-dependent as it is necessary for computing n-point functions by differentiating Γ k [φ ]. The running inverse propagator is given by (2) 2 Γk (x − y) = δ Γ k∕δφ(x) δφ(y), for example. Here a new potential cutoff scale enters the game: the momentum q dual to the distance x − y. When it serves as the operative IR cutoff in the denominator of the multiply differentiated FRGE, the running of Γ (2)(q) k, the Fourier transform of Γ (2)(x − y ) k, stops once k2 is smaller than 2 2 kdec = q. Hence (2) (2) √ -- Γ k (q) ≈ Γk (q)|k= q2 for 2 2 k ≲ q, provided no other physical scales intervene. As a result, if one allows for a running Z-factor in the truncation (C.22View Equation) one predicts a propagator of the type --- [Z (∘ q2)q2]−1 in the standard effective action. Note that generically it corresponds to a nonlocal term ∫ √ ---2- 2 ∝ φZ ( − ∂ )∂ φ in Γ, even though the truncation ansatz was local.

In the context of the effective average action formalism for gravity this kind of reasoning [135134] also underlies the evaluation of the UV behavior of the propagators in the “anomalous dimension argument” of Section 2.4. If η N is approximately constant, the graviton Z-factor varies as Z (k) ∼ k −ηN N, and the corresponding propagator ∘ -2 2 −1 [Z( q)q ] is proportional to 2 −1+ηN∕2 (q ) in momentum space and to √ --- ( x2)2−d−ηN in position space.

In the literature similar arguments have been used for the “renormalization group improvement” of cosmological [373818328] and black hole spacetimes [363540] on the basis of the effective average action (see also [189191190] for a discussion of different improvement schemes).

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