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4.1 The effective average action for gravity and its FRGE

The effective average action is a scale dependent variant Γ k of the usual effective action Γ, modified by a mode-cutoff k, such that Γ k can be interpreted as describing an ‘effective field theory at scale k’. For non-gauge theories a self-contained summary of this formalism can be found in Appendix C. In the application to gauge theories and gravity two conceptual problems occur.

First the standard effective action is not a gauge invariant functional of its argument. For example if in a Yang–Mills theory one gauge-fixes the functional integral with an ordinary gauge fixing condition like ∂ μAaμ = 0, couples the Yang–Mills field Aaμ to a source, and constructs the ordinary effective action, the resulting functional Γ [Aa ] μ is not invariant under the gauge transformations of a A μ. Although physical quantities extracted from a Γ [A μ] are expected to be gauge invariant, the noninvariance is cumbersome for renormalization purposes. The second problem is related to the fact that in a gauge theory a “coarse graining” based on a naive Fourier decomposition of a Aμ(x ) is not gauge covariant and hence not physical. In fact, if one were to gauge transform a slowly varying Aaμ(x) with a parameter function ω(x ) with a fast x-variation, a gauge field with a fast x-variation would arise, which however still describes the same physics.

Both problems can be overcome by using the background field formalism. The background effective action generally is a gauge invariant functional of its argument (see Appendix B). The second problem is overcome by using the spectrum of a covariant differential operator built from the background field configuration to discrimate between slow modes (small eigenvalues) and fast modes (large eigenvalues) [187Jump To The Next Citation Point]. This sacrifices to some extent the intuition of a spatial coarse graining, but it produces a gauge invariant separation of modes. Applied to a non-gauge theory it amounts to expanding the field in terms of eigenfunctions of the (positive) operator − ∂2 and declaring its eigenmodes ‘long’ or ‘short’ wavelength depending on whether the corresponding p2 is smaller or larger than a given k2.

This is the strategy adopted to define the effective average action for gravity [179Jump To The Next Citation Point]. In short: The effective average action for gravity is a variant of the background effective action Γ [⟨fαβ⟩,σα,σ¯α;¯gαβ] described in Appendix B (see Equations (B.48View Equation, B.51View Equation)), where the bare action is modified by mode cutoff terms as in Appendix C, but with the mode cutoff defined via the spectrum of a covariant differential operator built from the background metric. For convenience we quickly recapitulate the main features of the background field technique here and then describe the modifications needed for the mode cutoff.

The initial bare action S [g ] is assumed to be a reparameterization invariant functional of the metric g = (gαβ)1≤α,β≤d. Infinitesimally the invariance reads 2 S [g + ℒvg ] = S[g] + O (v ), where ℒvg is the Lie derivative of g with respect to the vector field γ v ∂γ. The metric g (later the integration variable in the functional integral) is decomposed into a background ¯g and a fluctuation f, i.e. gαβ = ¯gαβ + fαβ. The fluctuation field fαβ is then taken as the dynamical variable over which the functional integral is performed; it is not assumed to be small in some sense, no expansion in powers of f is implied by the split. Note however that this linear split does not have a geometrical meaning in the space of geometries. The symmetry variation gα β ↦→ gαβ + ℒvg αβ can be decomposed in two different ways

f ↦→ f + ℒ (¯g + f) , ¯g ↦→ ¯g , (4.1 ) αβ αβ v αβ αβ αβ fαβ ↦→ fαβ + ℒvfαβ, ¯gαβ ↦→ ¯gαβ + ℒvg¯α β. (4.2 )
We shall refer to the first one as “genuine gauge transformations” and to the second one as the “background gauge transformations”. The background effective action α Γ [⟨fαβ⟩,σ ,¯σ α;¯gαβ] is a functional of the expectation value ⟨fαβ⟩ of the fluctuation variable, the background metric ¯gαβ, and the expectation values of the ghost fields σα = ⟨C α⟩, ¯σ α = ⟨ ¯Cα⟩. Importantly Γ [⟨fαβ⟩,σ α,¯σα;¯gαβ] is invariant under the background field transformations (4.2View Equation). So far one should think of the background geometry as being prescribed but of generic form; eventually it is adjusted self-consistently by a condition involving the full effective action (see Equation (2.48View Equation) and Appendix B).

In the next step the initial bare action should be replaced by one involving a mode cutoff term. In the background field technique the mode cutoff should be done in a way that preserves the invariance under the background gauge transformations (4.2View Equation). We now first present the steps leading to the scale dependent effective average action Γ k[⟨gαβ ⟩ − ¯gαβ,σ α,¯σα;¯gαβ] in some detail and then present the FRGE for it. The functional integrals occuring are largely formal; for definiteness we consider the Euclidean variant where the integral over Riemannian geometries is intended. The precise definition of the generating functionals is not essential here, as they mainly serve to arrive at the gravitional FRGE. The latter provides a novel tool for investigating the gravitational renormalization flow.

We begin by introducing a scale dependent variant Wk of the generating functional of the connected Greens functions. The cutoff scale is again denoted by k, it has unit mass dimension, and no physics interpretation off hand. The defining relation for Wk reads

{ } ∫ grav ∫ √ -- αβ Wk [J,j,¯j; ¯g] = 𝒟μj,¯j,k[f] exp − S[¯g + f] − C k [f;¯g] + dx ¯gJ fαβ . (4.3 )
Here the measure 𝒟 μj,¯j,k[f] differs from the naive one, 𝒟f αβ, by gauge fixing terms and an integration over ghost fields C α, C¯α, where the action for the latter is again modified by a mode-cutoff:
∫ { 1 ∫ √ -- 1 ∫ ∂Q β 𝒟 μj,¯j,k[f ] = 𝒟f αβ 𝒟C α𝒟 C¯α exp − --- dx ¯g¯gαβQ αQ β + -- dxC¯α¯gαβ ----ℒC (¯g + f)γδ 2α ∫ κ } ∂fγδ gh ¯ √ -- α α ¯ − C k [C, C ] + dx ¯g[¯jαC + j Cα ] . (4.4 )
The first term in the exponent is the gauge fixing term. The gauge fixing condition Q α = Q α[¯g;f] ≈ 0 must be invariant under Equation (4.2View Equation), for the moment we may leave it unspecified. The second term is the Faddeev–Popov action for the ghosts obtained in the usual way: One applies a genuine gauge transformation (4.1View Equation) to Qα and replaces the parameter α v by the ghost field α C. The integral over α C and ¯Cα then exponentiates the Fadeev–Popov determinant det[δQ α∕δvβ]. This gauge fixing procedure has a somewhat perturbative flavor; large scale aspects of the space of geometries are not adequately taken into account. The terms Cgrav k and Cgh k implement the mode cutoff in the gravity and the ghost sector, respectively. We shall specify them shortly. Finally we coupled in Equation (4.4View Equation) the ghosts to sources α j, j¯α for later use.

The construction of the effective average action now parallels that in the scalar case. We quickly run through the relevant steps. The Legendre transform of W k at fixed ¯g αβ is

{ ∫ } √ -( αβ α α) ^Γ k[f¯,σ,¯σ;¯g] = sup dx ¯g ¯fαβJ + σ ¯jα + ¯σ αj − Wk [J,j,¯j;¯g] . (4.5 ) J,j,¯j
As usual, if Wk is differentiable with respect to the sources, the extremizing source configurations J α∗β, ¯j∗,α, jα ∗ allow one to interpret ¯fαβ, σα, ¯σα as the expectation values of fαβ, Cα, C¯α via
1 δWk α α 1 δWk 1 δWk ¯fαβ = ⟨fαβ⟩ = √-¯g---αβ, σ = ⟨C ⟩ = √-¯g-δj--, ¯σα = ⟨C¯α⟩ = √-¯g-δjα-. (4.6 ) δJ∗ ∗,α ∗
Note that the expectation values defined through Equation (4.3View Equation) are in general both k-dependent and source dependent. In Equation (4.6View Equation), by construction, the k-dependence carried by Wk cancels that carried by the extremizing source. Concretely the extremizing sources are constructed by assuming that Wk has a series expansion in powers of the sources with ¯g-dependent coefficients; formal inversion of the series then gives a k-dependent Jα∗β[¯f,σ, ¯σ;¯g] with the property that Jα∗β[0,σ,¯σ; ¯g] = 0, and similarly for the ghost sources. The formal effective field equations dual to Equation (4.6View Equation) read
-δ--^Γ [f¯,σ,¯σ;¯g] = Jαβ [f¯,σ,¯σ;¯g], δ ¯fαβ k ∗ --δ-^ ¯ α ¯ (4.7 ) δ ¯σαΓ k[f ,σ,¯σ;¯g] = j∗[f,σ,¯σ; ¯g], δ δ-σα^Γ k[f¯,σ,¯σ;¯g] = ¯j∗,α[¯f,σ,σ¯; ¯g].
As in the scalar case the effective average action differs from ^Γ k by the cutoff action with the expectation value fields inserted,
grav gh Γ k[f¯,σ,¯σ;¯g] := Γ^k[f¯,σ,¯σ;¯g ] − C k [¯f;¯g] − C k [σ,¯σ;¯g]. (4.8 )
Sometimes it is convenient to introduce ⟨g ⟩ := ¯g + ¯f αβ αβ αβ, which is the expectation value of the original ‘quantum’ metric gαβ = ¯gαβ + fαβ, and to regard Γ k as a functional of ⟨gαβ⟩ rather than ¯ fαβ, i.e. Γ k[⟨gαβ⟩,¯gαβ,σ α,¯σα] := Γ k[⟨gαβ⟩ − ¯gαβ,σ, ¯σ;¯g].

Usually one is not interested in correlation functions involving Faddeev–Popov ghosts and it is sufficient to know the reduced functional

¯Γ k[g] := Γ k[0,0,0;g ] ≡ Γ k[g,g, 0,0]. (4.9 )
As indicated we shall simply write g αβ for its argument ⟨g ⟩ = ¯g αβ αβ.

The precise form of the gauge condition Qα [¯g;f ] is inessential, only the invariance under Equation (4.2View Equation) is important. It ensures that the associated ghost action is invariant under Equation (4.2View Equation) and δCα = ℒvC α, δ ¯Cα = ℒv ¯C α. We shall ignore the problem of the global existence of gauge slices (“Gribov copies”), in accordance with the formal nature of the construction. For later reference let us briefly describe the most widely used gauge condition, the “background harmonic gauge” which reads

[ ] : √ -- ¯β 1- γγ′ ′ Q α[¯g,f ]= 2κ∇ fαβ − 2¯gαβ ¯g f γγ . (4.10 )
The covariant derivative ¯∇ α involves the Christoffel symbols ¯Γ γ αβ of the background metric. Note that Q α of Equation (4.10View Equation) is linear in the quantum field f αβ. On a flat background with ¯g = η αβ αβ the condition Q α = 0 reduces to the familiar harmonic gauge condition, β 1 β ∂ fβα = 2∂αfβ. In Equation (4.10View Equation) κ is an arbitrary constant with the dimension of a mass. We shall set κ = (2gN )−1∕2 and interpret gN = 16πG as the bare Newton constant. The ghost action for the gauge condition (4.10View Equation) is
√--∫ √ -- − 2 dx ¯gC¯αℳ [¯g + f, ¯g]α β C β, (4.11 ) with ℳ [g,¯g]αβ = ¯gα ρg¯σλ¯∇ λ(g ρβ∇ σ + gσβ∇ ρ) − ¯gρσ¯gαλ¯∇ λgσβ∇ ρ, (4.12 )
where ∇ α and ¯∇ α are the covariant derivatives associated with g αβ (as a short for ¯g + f αβ αβ) and ¯gαβ, respectively.

The last ingredient in Equations (4.3View Equation, 4.4View Equation) to be specified are the mode cutoff terms, not present in the usual background effective action. Their precise form is arbitrary to some extent. Naturally they will be taken quadratic in the respective fields, with a kernel which is covariant under background gauge transformations. These requirements are met if

2∫ Cgrav[f;¯g] = κ-- dx √ ¯gf ℛ [¯g]αβγδf , k 2 αβ k γδ √ --∫ √ -- (4.13 ) Cgkh[C, ¯C;¯g] = 2 dx ¯gC¯αℛghk [¯g]C α,
and the kernels ℛk [g¯], ℛgh [¯g] k transform covariantly under ¯gαβ ↦→ ¯gαβ + ℒv¯gαβ. In addition they should effectively suppress covariant ‘momentum modes’ with ‘momenta’ 2 2 p < k. As mentioned earlier one way of defining such a covariant scale is via the spectrum of a covariant differential operator. Concretely the following choice will be used.

Consider the Laplacian αβ ¯ ¯ Δ ¯g := ¯g ∇ α∇ β of the Riemannian background metric ¯g with ¯ ∇ being its torsionfree connection. We assume ¯g to be such that − Δ ¯g has a non-negative spectrum and a complete set of (tensorial) eigenfunctions. The spectral values λ = λ[¯g] of − Δ ¯g will then be functionals of ¯g and one can choose ℛ [¯g] k and ℛgh [¯g] k such that only eigenmodes with spectral values λ[¯g] ≫ k2 (2 being the mass dimension of the operator) enter the fαβ functional integral unsuppressed. Here one should think of the fαβ functional integral as being replaced by one over the (complete system of) eigenfunctions of − Δ ¯g, for a fixed ¯g. Concretely, for ℛk [g¯] and ℛgkh[¯g] we take expressions of the form

αβγδ αβγδ 2 (0) 2 gh gh 2 (0) 2 ℛk [¯g] = 𝒵 k [¯g]k ℛ (− Δ ¯g∕k ), ℛ k [¯g] = 𝒵 k k ℛ (− Δ ¯g∕k ). (4.14 )
As indicated in Equation (4.14View Equation) the prefactors 𝒵k are different for the gravitational and the ghost cutoff. For the ghosts 𝒵gh k is a pure number, whereas for the metric fluctuation 𝒵 αβγδ[¯g] k is a tensor constructed from the background metric g¯αβ. We shall discuss the choice of these prefactors later on.

The essential ingredient in Equation (4.14View Equation) is a function ℛ (0) : ℝ+ → ℝ+ interpolating smoothly between ℛ (0)(0) = 1 and ℛ (0)(∞ ) = 0; for example

(0) −1 ℛ (u) = u [exp (u ) − 1 ] . (4.15 )
Its argument u should be identified with the weighted spectral values λ[¯g]∕k2 of − Δ ¯g. One readily sees that then the exponentials in Equation (4.13View Equation) have the desired effect: They effectively suppress eigenmodes of − Δ ¯g with spectral values much smaller than 2 k, while modes with λ [¯g] large compared to k2 are unaffected. This also illustrates that a mode suppression can be defined covariantly using the background field formalism.

This concludes the definition of the effective average action and its various specializations. We now present its key properties.

4.1.1 Properties of the effective average action

  1. The effective average action is invariant under background field diffeomorphisms
    Γ k[Φ + ℒvΦ ] = Γ k[Φ ], Φ := {⟨gαβ⟩,¯gαβ,σ α,¯σα} , (4.16)
    where all its arguments transform as tensors of the corresponding rank. This is a direct consequence of the corresponding property of Wk in Equation (4.3View Equation)
    { } Wk [𝒥 + ℒv𝒥 ] = Wk [𝒥 ], 𝒥 := Jαβ, jα, ¯jα, ¯gαβ , (4.17)
    where ℒv is the Lie derivative of the respective tensor type. Here Equation (4.16View Equation) is obtained as in Equation (B.47View Equation) of Appendix B, where the background covariance of the mode cutoff terms (4.13View Equation) is essential. Further the measure 𝒟f αβ is assumed to be diffeomorphism invariant.
  2. It satisfies the functional integro-differential equation
    { } ∫ { exp − Γ k[f¯,σ,¯σ;¯g] = 𝒟f 𝒟C 𝒟 C¯ exp − Stot[f,C, ¯C;g¯] − Ck [f − f¯,C − σ, ¯C − ¯σ] ∫ √ -[ δΓ k δΓ k δΓ k] } + dx ¯g (fαβ − f¯α β)-¯---+ (C α − σα)---α + ( ¯Cα − ¯σα)---- , δfαβ δσ δ¯σα (4.18)
    where Stot := S + Sgf + Sgh (with Sgf and Sgh minus the first two terms in the exponent of Equation (4.4View Equation) and grav gh Ck := Ck + C k.
  3. The k-dependence of the effective average action is governed by an exact FRGE. Following the same lines as in the scalar case one arrives at [179Jump To The Next Citation Point]
    [( )− 1( ) ] k ∂ Γ [¯f,σ,σ¯; ¯g] = 1-Tr Γ (2)+ ℛ^ k ∂ ^ℛ k k 2 k k f¯f¯ k k ¯f ¯f 1 [{ ( )−1 ( ) −1} ( ) ] − -Tr Γ (2k)+ ^ℛk − Γ (2k)+ ^ℛk k ∂k ^ℛk . (4.19) 2 ¯σσ σ¯σ ¯σσ
    Here Γ (2) k denotes the Hessian of Γ k with respect to the dynamical fields f¯, σ, ¯σ at fixed ¯g. It is a block matrix labeled by the fields : ¯ ϕi = {fαβ, α σ ,¯σα },
    2 ^Γ (2)ij(x,y ) := ∘----1----------δ-^Γ k---. (4.20) k ¯g(x )g¯(y)δϕi (x )δϕj(y)
    (In the ghost sector the derivatives are understood as left derivatives.) Likewise, ℛ^k is a block matrix with entries (ℛ^k )α¯β¯γδ:= κ2ℛgrav[¯g]αβγδ ff k and √ -- ^ℛ ¯σσ = 2ℛgh [¯g] k. Performing the trace in the position representation it includes an integration ∫ ∘ ----- dx ¯g(x ) involving the background volume element. For any cutoff which is qualitatively similar to Equation (4.14View Equation, 4.15View Equation) the traces on the right-hand-side of Equation (4.19View Equation) are well convergent, both in the infrared and in the ultraviolet. By virtue of the factor k∂ ^ℛ k k, the dominant contributions come from a narrow band of generalized momenta centered around k. Large momenta are exponentially suppressed.

    The conceptual status and the use of the gravitational FRGE (4.20View Equation) is the same in the scalar case discussed in Section 2.2. Its perturbative expansion should reproduce the traditional non-renormalizable cutoff dependencies starting from two-loops. In the context of the asymptotic safety scenario the hypothesis at stake is that in an exact treatment of the equation the cutoff dependencies entering through the initial data get reshuffled in a way compatible with asymptotic safety. The Criterion (FRGC1) for the existence of a genuine continuuum limit discussed in Section 2.3 also applies to Equation (4.20View Equation). In brief, provided a global solution of the FRGE (4.20View Equation) can be found (one which exists for all 0 ≤ k ≤ ∞), it can reasonably be identified with a renormalized effective average action lim Λ→∞ Γ Λ,k constructed by other means. The intricacies of the renormalization process have been shifted to the search for fine-tuned initial functionals for which a global solution of Equation (4.20View Equation) exists. For such a global solution limk →0Γ k then is the full quantum effective action and limk → ∞ Γ k = S∗ is the fixed point action. As already noted in Section 2.3 the appropriate positivity requirement (FRGC2) remains to be formulated; one aspect of it concerns the choice of 𝒵 k factors in Equation (4.13View Equation) and will be discussed below.

    The background gauge invariance of Γ k expressed in Equation (4.16View Equation) is of great practical importance. It ensures that if the initial functional does not contain non-invariant terms, the flow implied by the above FRGE will not generate such terms. In contrast locality is not preserved of course; even if the initial functional is local the flow generates all sorts of terms, both local and nonlocal, compatible with the symmetries.

    For the derivation of the flow equation it is important that the cutoff functionals in Equation (4.13View Equation) are quadratic in the fluctuation fields; only then a flow equation arises which contains only second functional derivatives of Γ k, but no higher ones. For example using a cutoff operator involving the Laplace operator of the full metric gαβ = ¯gα β + fα β would result in prohibitively complicated flow equations which could hardly be used for practical computations.

    For most purposes the reduced effective average action (4.9View Equation) is suffient and it is likewise background invariant, ¯Γ k[g + ℒvg] = ¯Γ k[g]. Unfortunately ¯Γ k[¯gαβ] does not satisfy an exact FRGE, basically because it contains too little information. The actual RG evolution has to be performed at the level of the functional Γ k[⟨g⟩,¯g,σ,σ¯]. Only after the evolution one may set ⟨g⟩ = ¯g, σ = 0,¯σ = 0. As a result, the actual theory space of Quantum Einstein Gravity in this setting consists of functionals of all four variables, ⟨gαβ ⟩,g¯αβ,σα,¯σα, subject to the invariance condition (4.9View Equation). Since Γ (2) k involves derivatives with respect to f¯ αβ at fixed ¯gαβ it is clear that the evolution cannot be formulated in terms of ¯ Γ k alone.

  4. Γ k[¯f,σ,σ¯; ¯g] approaches for k → 0 the background effective action of Appendix B, since grav ℛ k, ℛghk vanish for k → 0. The k → ∞ limit can be infered from Equation (4.18View Equation) by the same reasoning as in the scalar case (see Appendix C). This gives
    kli→m∞ Γ k[¯f,σ,¯σ; ¯g] = Stot[¯f,σ,¯σ; ¯g]. (4.21)
    Note that the bare initial functional Γ k includes the gauge fixing and ghost actions. At the level of the functional ¯Γ k[g] Equation (4.21View Equation) reduces to limk→ ∞ ¯Γ k[g] = S[g].
  5. The effective average action satisfies a functional BRST Ward identity which reflects the invariance of Stot under the BRST transformations
    δεfαβ = εκ− 2ℒC (¯gαβ + fαβ), δε¯gαβ = 0, (4.22) δεC α = εκ− 2C β∂ βCα, δεC ¯α = ε(ακ)− 1Qα.
    Here ε is an anti-commuting parameter. Since the mode cutoff action Ck is not BRST invariant, the Ward identity differs from the standard one by terms involving grav ℛ k, gh ℛ k. For the explicit form of the identity we refer to [179Jump To The Next Citation Point].
  6. Initially the vertex (or 1-PI Greens) functions are given by multiple functional derivatives of Γ k[¯f,σ,σ¯; ¯g] with respect to f¯,σ, ¯σ at fixed ¯g and setting
    ¯ α α ¯ fαβ = ⟨fα β⟩ = 0, σ = ⟨C ⟩ = 0, ¯σα = ⟨C α⟩ = 0 (4.23)
    after differentiation. The resulting multi-point functions Γ (nk)(x1,...,xn;g ) are k-dependent functionals of the (k-independent) ⟨gαβ⟩ = ¯gαβ. For extremizing sources obtained by formal series inversion the condition (4.23View Equation) automatically switches off the sources in Equation (4.7View Equation); for the ghosts this is consistent with Γ k having ghost number zero. Note that the one-point function (1) Γk (x1;g ) ≡ 0 vanishes identically.

    An equivalent set of vertex functions should in analogy to the Yang–Mills case [68Jump To The Next Citation Point4267] be obtained by differentiating Γ k[g¯, σ,¯σ] := Γ k[0,σ, ¯σ;¯g] ≡ Γ k[¯g,¯g,σ,¯σ ] with respect to ¯g. Specifically for σ α = ¯σα = 0 one gets multipoint functions ¯Γ (n)(x1,...,xn ) k, with the shorthand (4.9View Equation). The solutions (ˇgk)αβ of

    δ¯Γ k ----[ˇgk] = 0 (4.24) δgαβ
    play an important role in the interpretation of the formalism (see Section4.2).

    The precise physics significance of the multipoint functions Γ (kn) and ¯Γ (kn) remains to be understood. One would expect them to be related to S-matrix elements on a self-consistent background but, for example, an understanding of the correct infrared degrees of freedom is missing.

This concludes our summary of the key properties of the gravitational effective average action. Before turning to applications of this formalism, we discuss the significance and the proper choice of the 𝒵k factors in Equation (4.14View Equation), which is one aspect of the positivity issue (FRGC2) of Section 2.2. The significance of these factors is best illustrated in the scalar case. As discussed in Appendix C in scalar theories with more than one field it is important that all fields are cut off at the same 2 k. This is achieved by a cutoff function of the form (C.21View Equation) where 𝒵k is in general a matrix in field space. In the sector of modes with inverse propagator Z (i)p2 + ... k the matrix 𝒵k is diagonal with entries 𝒵k = Z(i) k. In a scalar field theory these 𝒵k factors are automatically positive and the flow equations in the various truncations are well-defined.

In gravity the situation may be more subtle. First, consider the case where φ is some normal mode of f¯αβ and that it is an eigenfunction of Γ (2k) with eigenvalue Z φkp2, where p2 is a positive eigenvalue of some covariant kinetic operator, typically of the form − ¯∇2 + R-terms. If Z φ> 0 k the situation is clear, and the rule discussed in the context of scalar theories applies: One chooses φ 𝒵 = Z k because this guarantees that for the low momentum modes the running inverse propagator Γ (2k)+ ℛk becomes Z φ(p2 + k2) k, exactly as it should be.

More tricky is the question how 𝒵k should be chosen if φ Z k is negative. If one continues to use 𝒵k = Zkφ, the evolution equation is perfectly well defined because the inverse propagator − |Zφk|(p2 + k2) never vanishes, and the traces of Equation (4.19View Equation) are not suffering from any infrared problems. In fact, if we write down the perturbative expansion for the functional trace, for instance, it is clear that all propagators are correctly cut off in the infrared, and that loop momenta smaller than k are suppressed. On the other hand, if we set φ 𝒵k = − Zk, then φ − |Z k|(p2 − k2) introduces a spurios singularity at p2 = k2, and the cutoff fails to make the theory infrared finite. This choice of 𝒵k is ruled out therefore. At first sight the choice 𝒵k = − Z φ k might have appeared more natural because only if 𝒵k > 0 the factor ∫ exp (− Ck ) ∼ exp(− φℛk φ ) is a damped exponentially which suppresses the low momentum modes in the usual way. For the other choice 𝒵k = +Z φ < 0 k the factor ∫ exp{ |ℛk |φ2 } is a growing exponential instead and, at least at first sight, seems to enhance rather than suppress the infrared modes. However, as suggested by the perturbative argument, this conclusion is too naive perhaps.

In all existing RG studies using this formalism the choice

𝒵 = +Z φ, for either sign of Z φ, (4.25 ) k k k
has been adopted, and there is little doubt that, within those necessarily truncated calculations, this is the correct procedure. Besides the perturbative considerations above there are various arguments of a more general nature which support Equation (4.25View Equation):

In fact, as we shall discuss in more detail later on, the to date best truncation used for the investigation of asymptotic safety (“R2 truncation”) has only positive 𝒵 k factors in the relevant regime. On the other hand, the simpler “Einstein–Hilbert truncation” has also negative 𝒵k’s. If one applies the rule (4.25View Equation) to it, the Einstein–Hilbert truncation produces almost the same results as the R2 truncations [131Jump To The Next Citation Point132Jump To The Next Citation Point]. This is a strong argument in favor of the 𝒵k = +Z φk rule.

In [133Jump To The Next Citation Point131Jump To The Next Citation Point] a slightly more general variant of the construction described here has been employed. In order to facilitate the calculation of the functional traces in the FRGE (4.19View Equation) it is helpful to employ a transverse-traceless (TT) decomposition of the metric: fαβ = f T + ∇¯αV β + ¯∇ βVα + ¯∇ α¯∇ βσ − d−1¯gαβ ¯∇2σ + d −1¯gαβφ = f^αβ + d− 1¯gαβφ αβ. Here fT αβ is a transverse traceless tensor, V α a transverse vector, and σ and φ are scalars. In this framework it is natural to formulate the cutoff in terms of the component fields appearing in the TT decomposition ∫ αβ ∫ Ck ∼ fTαβℛkf T + V αℛkV α + .... This cutoff is referred to as a cutoff of “type B”, in contradistinction to the “type A” cutoff described above, ∫ Ck ∼ fαβℛkf αβ. Since covariant derivatives do not commute, the two cutoffs are not exactly equal even if they contain the same shape function ℛ (0). Thus, comparing type A and type B cutoffs is an additional possibility for checking scheme (in)dependence [133Jump To The Next Citation Point131Jump To The Next Citation Point].

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