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2.2 Functional flow equations and UV renormalization

The technique of functional renormalization group equations (FRGEs) does not rely on a perturbative expansion and has been widely used for the computation of critical exponents and the flow of generalized couplings. For a systematic exposition of this technique and its applications we refer to the reviews [146Jump To The Next Citation Point21Jump To The Next Citation Point166Jump To The Next Citation Point229Jump To The Next Citation Point29Jump To The Next Citation Point]. Here we shall mainly use the effective average action Γ k and its ‘exact’ FRGE. We refer to Appendix C for a summary of this formulation, and discuss in this section how the UV renormalization problem presents itself in an FRGE [157Jump To The Next Citation Point].

In typical applications of the FRG the ultraviolet renormalization problem does not have to be addressed. In the context of the asymptotic safety scenario this is different. By definition the perturbative series in a field theory based on an asymptotically safe functional measure has a dependence on the UV cutoff which is not strictly renormalizable (see Section 1.3). The perturbative expansion of an FRGE must reproduce the structure of these divergencies. On the other hand in an exact treatment or based on different approximation techniques a reshuffling of the cutoff dependence is meant to occur which allows for a genuine continuum limit. We therefore outline here how the UV renormalization problem manifests itself in the framework of the functional flow equations. The goal will be to formulate a criterion for the plausible existence of a genuine continuum limit in parallel to the one above based on perturbative indicators.

Again we illustrate the relevant issues for a scalar quantum field theory on flat space. For definiteness we consider here the flow equation for the effective average action Γ Λ,k[φ], for other types of FRGEs the discussion is similar though. The effective average action interpolates between the bare action SΛ [φ] and the above, initially regulated, effective action Γ Λ, according to

k→Λ k→0 SΛ[φ] ← − Γ Λ,k[φ] −→ Γ Λ[φ]. (2.7 )
Roughly speaking one should think of Γ Λ,k[φ] as the conventional effective action but with only the momentum modes in the range 2 2 2 k < p < Λ integrated out. The k → Λ limit in Equation (2.7View Equation) will in fact differ from S Λ by a 1-loop determinant ln det[S(2) + ℛ Λ] Λ (see Appendix C.2). For the following discussion the difference is inessential and for (notational) simplicity we will identify Γ Λ,Λ with SΛ. Equation (2.7View Equation) also presupposes that for fixed UV cutoff Λ the limit k → 0 exists, which for theories with massless degrees of freedom is nontrivial.

The conventional effective action obeys a well-known functional integro-differential equation which implicitly defines it (see Equation (B.8View Equation) below). Its counterpart for Γ Λ,k[φ] reads

∫ { ∫ } exp {− Γ [φ ]} = [𝒟 χ ] exp − S [χ] + dx (χ − φ )(x )δΓ Λ,k[φ-] , (2.8 ) Λ,k Λ,k Λ δφ(x)
where the functional measure [𝒟 χ ]Λ,k includes mostly momentum modes in the range 2 2 2 k < p < Λ. This can be done by multiplying the kinematical measure by a suitable mode suppression factor
[𝒟 χ] = 𝒟χ exp {− C [χ − φ ]}, (2.9 ) Λ,k Λ,k
with a suitable quadratic form CΛ,k. From Equation (2.8View Equation) one can also directly verify the alternative characterization (1.12View Equation).

The precise form of the mode suppression is inessential. In the following we outline a variant which is technically convenient. Here C Λ,k is a quadratic form in the fields defined in terms of a kernel ℛ Λ,k chosen such that both ℛ Λ,k and k ∂kℛ Λ,k define integral operators of trace-class on the function space considered. We write ∫ [ℛ Λ,kχ](x) := dy ℛ Λ,k(x,y )χ (y) for the integral operator and Tr [ℛ Λ,k] := ∫ dxℛ Λ,k(x,x) < ∞ for its trace. The other properties of the kernel are best described in Fourier space, where ℛk,Λ acts as ∫ -dq- [ℛ Λ,k^χ](p) = (2π)d ℛ Λ,k(p,q) ^χ (q), with ∫ ^χ (p) = dx χ (x)exp(− ipx), the Fourier transform of χ and similarly for the kernel (where we omit the hat for notational simplicity). The UV cutoff 0 ≤ p2 < Λ renders Euclidean momentum space compact and Mercer’s theorem then provides simple sufficient conditions for an integral operator to be trace-class [157Jump To The Next Citation Point]. We thus take the kernel ℛ Λ,k(p,q) to be smooth, symmetric in p,q, and such that

∫ ∫ 1 dp dq ∗ 1 ∗ CΛ,k[χ] := 2 (2π)d-(2π)d-^χ(q) ℛ Λ,k(p, q) ^χ(p) = 2 dx χ(x) [ℛΛ,kχ](x) ≥ 0 (2.10 )
for all continuous functions χ (similarly for k ∂kℛ Λ,k(p,q)). The trace-class condition is then satisfied and one can adjust the other features of the kernel to account for the mode suppression. These features are arbitrary to some extent; what matters is the limiting behavior for p2,q2 ≫ k2 and (with foresight) Λ → ∞. We refer to Appendix C for more details.

The presence of the extra scale k allows one to convert Equation (2.8View Equation) into a functional differential equation [228Jump To The Next Citation Point229Jump To The Next Citation Point29Jump To The Next Citation Point],

[ ] ∂ 1 ∂ ( (2) ) −1 k---Γ Λ,k[φ] = --Tr k---ℛ Λ,k Γ Λ,k[φ] + ℛ Λ,k (2.11 ) ∂k 2 ∫ ∂k ( ) 1- -dq1---dq2-- (2) −1 = 2 (2 π)d (2 π)dk∂kℛ Λ,k(q1,q2) Γ Λ,k[φ] + ℛ Λ,k (q2,q1),
known as the functional renormalization group equation (FRGE) for the effective average action. For convenience we include a quick derivation of Equation (2.11View Equation) in Appendix C. In the second line of Equation (2.11View Equation) we spelled out the trace using that k∂kℛ Λ,k is trace-class. Further (2) Γ Λ,k[φ] is the integral operator whose kernel is the Hessian of the effective average action, i.e. Γ (2)(x,y ) := δ2Γ Λ,k[φ ]∕(δφ (x)δφ(y )) Λ,k, and ℛ Λ,k is the integral operator in Equation (2.8View Equation).

For finite cutoffs (Λ, k) the trace of the right-hand-side of Equation (2.11View Equation) will exist as the potentially problematic high momentum parts are cut off. In slightly more technical terms, since the product of a trace-class operator with a bounded operator is again trace-class, the trace in Equation (2.11View Equation) is finite as long as the inverse of Γ (2)[φ ] + ℛ Λ,k Λ,k defines a bounded operator. For finite UV cutoff one sees from the momentum space version of Equation (B.2View Equation) in Section 3.4 that this will normally be the case. The trace-class property of the mode cutoff operator (for which Equation (2.10View Equation) is a sufficient condition) also ensures that the trace in Equation (2.11View Equation) can be evaluated in any basis, the momentum space variant displayed in the second line is just one convenient choice.

Importantly the FRGE (2.11View Equation) is independent of the bare action SΛ, which enters only via the initial condition Γ Λ,Λ = S Λ (for large Λ). In the FRGE approach the calculation of the functional integral for Γ Λ,k is replaced by the task of integrating this RG equation from k = Λ, where the initial condition Γ Λ,Λ = S Λ is imposed, down to k = 0, where the effective average action equals the ordinary effective action Γ Λ.

All this has been for a fixed UV cutoff Λ. The removal of the cutoff is of course the central theme of UV renormalization. In the FRG formulation one has to distinguish between two aspects: first, removal of the explicit Λ dependence in the trace on the right-hand-side of Equation (2.11View Equation), and second removal of the UV cutoff in Γ Λ,k itself, which was needed in order to make the original functional integral well-defined.

The first aspect is unproblematic: The trace is manifestly finite as long as the inverse of (2) ΓΛ,k[φ ] + ℛ Λ,k defines a bounded operator. If now Γ (2)[φ] Λ,k is independently known to have a finite and nontrivial limit as Λ → ∞, the explicit Λ dependence carried by the ℛ Λ,k term is harmless and the trace always exists. Roughly this is because the derivative kernel k∂kℛ Λ,k has support mostly on a thin shell around p2 ≈ k2, so that the (potentially problematic) large p behavior of the other factor is irrelevant (cf. Appendix C.2).

The second aspect of course relates to the traditional UV renormalization problem. Since Γ Λ,k came from a regularized functional integral it will develop the usual UV divergencies as one attempts to send Λ to infinity. The remedy is to carefully adjust the bare action S Λ[φ ] – that is, the initial condition for the FRGE (2.11View Equation) – in such a way that functional integral – viz. the solution of the FRGE – is asymptotically independent of Λ. Concretely this could be done by fine-tuning the way how the parameters u α(Λ) in the expansion ∑ SΛ [χ ] = αu α(Λ)Pα [χ ] depends on Λ. However the FRGE method in itself provides no means to find the proper initial functional S Λ[χ]. Identification of the fine-tuned SΛ [χ ] lies at the core of the UV renormalization problem, irrespective of whether Γ Λ,k is defined directly via the functional integral or via the FRGE. Beyond perturbation theory the only known techniques to identify the proper S Λ start directly from the functional integral and are ‘constructive’ in spirit (see [195Jump To The Next Citation Point48]). Unfortunately four-dimensional quantum field theories of interest are still beyond constructive control.

One may also ask whether perhaps the cutoff-dependent FRGE (2.11View Equation) itself can be used to show that a limit lim Λ→ ∞ Γ Λ,k[φ ] exists. Indeed using other FRGEs and a perturbative ansatz for the solution has lead to economic proofs of perturbative renormalizability, i.e. of the existence of a formal continuum limit in the sense of Criterion (PTC1) discussed before (see [200123Jump To The Next Citation Point]). Unfortunately so far this could not be extended to construct a nonperturbative continuum limit of fully fledged quantum field theories (see [145] for a recent review of such constructive uses of FRGEs). For the time being one has to be content with the following if then statement:

If there exists a sequence of initial actions SnΛ0[χ], n ∈ ℕ, such that the solution Γ nΛ0,k[φ] of the FRGE (2.11View Equation) remains finite as n → ∞, then the limit Γ k[φ] := limn → ∞ Γ nΛ0,k[φ ] has to obey the cut-off independent FRGE

[ ( )−1] k-∂-Γ k[φ] = 1-Tr k-∂-ℛk,∞ Γ (2)[φ] + ℛk,∞ . (2.12 ) ∂k 2 ∂k k
Conversely, under the above premise, this equation should have at least one solution with a finite limit limk → ∞ Γ k[φ]. This limit can now be identified with the renormalized fixed point action S∗[χ ]. It is renormalized because by construction the cutoff dependencies have been eaten up by the ones produced by the trace in Equation (2.11View Equation). It can be identified with a fixed point action because lowering k amounts to coarse graining, and S ∗[χ ] is the ‘inverse limit’ of a sequence of such coarse graining steps.

So far the positivity or unitarity requirement has not been discussed. From the (Osterwalder–Schrader or Wightman) reconstruction theorems it is known how the unitarity of a quantum field theory on a flat spacetime translates into nonlinear conditions on the multipoint functions. Since the latter can be expressed in terms of the functional derivatives of Γ k, unitarity can in principle be tested retroactively, and is expected to hold only in the limit k → 0. Unfortunately this is a very indirect and retroactive criterion. One of the roles of the bare action S Λ[χ] = Γ Λ,Λ[χ ] is to encode properties which are likely to ensure the desired properties of limk →0 Γ k[φ]. In theories with massless degrees of freedom the k → 0 limit is nontrivial and the problem of gaining computational control over the infrared physics should be separated from the UV aspects of the continuum limit as much as possible. However the k → 0 limit is essential to probe stability/positivity/unitarity.

One aspect of positivity is the convexity of the effective action. The functional equations (2.11View Equation, 2.12View Equation) do in itself “not know” that Γ k is the Legendre transform of a convex functional and hence must be itself convex. Convexity must therefore enter through the inital data and it will also put constraints on the choice of the mode cutoffs. Good mode cutoffs are characterized by the fact that Γ (2) + ℛk k has positive spectral values for all k (cf. Equation (C.14View Equation)). If no blow-up occurs in the flow the limit (2) limk →0Γ k will then also have non-negative spectrum. Of course this presupposes again that the proper initial conditions have been identified and the role of the bare action is as above.

For flat space quantum field theories one expects that S Λ[χ ] must be local, i.e. a differential polynomial of finite order in the fields so as to end up with an effective action limk → ∞ lim Λ→ ∞ Γ Λ,k[φ ] describing a local/microcausal unitary quantum field theory.

For convenient reference we summarize these conclusions in the following criterion:

Criterion (Continuum limit in the functional RG approach):
 
(FRGC1) A solution of the cutoff independent FRGE (2.12View Equation) which exists globally in k (for all 0 ≤ k ≤ ∞) can reasonably be identified with the continuum limit of the effective average action lim Λ→ ∞ Γ Λ,k[φ] constructed by other means. For such a solution limk →0Γ k[φ ] is the full quantum effective action and limk → ∞ Γ k[φ] = S∗[φ] is the fixed point action.

 
(FRGC2) For a unitary relativistic quantum field theory positivity/unitarity must be tested retroactively from the functional derivatives of lim Γ [φ ] k→0 k.

We add some comments:

Since the FRGE (2.12View Equation) is a differential equation in k, an initial functional Γ [φ ] initial has to be specified for some 0 < kinitial ≤ ∞, to generate a local solution near k = kinitial. The point is that for ‘almost all’ choices of Γ initial[φ ] the local solution cannot be extended to all values of k. Finding the rare initial functionals for which this is possible is the FRGE counterpart of the UV renormalization problem. The existence of the k → 0 limit is itself not part of the UV problem; in conventional quantum field theories the k → 0 limit is however essential to probe unitarity/positivity/stability.

It is presently not known whether the above criterion can be converted into a theorem. Suppose for a quantum field theory on the lattice (with lattice spacing Λ −1) the effective action Γ latt[φ] Λ,k has been constructed nonperturbatively from a transfer operator satisfying reflection positivity and that a continuum limit latt lim Λ→ ∞ ΓΛ,k is assumed to exist. Does it coincide with a solution Γ k[φ] of Equation (2.12View Equation) satisfying the Criteria (FRGC1) and (FRGC2)? Note that this is ‘only’ a matter of controlling the limit, for finite Λ also Γ laΛt,tk will satisfy the flow equation (2.11View Equation).

For an application to quantum gravity one will initially only ask for Criterion (FRGC1), perhaps with even only a partial understanding of the k → 0 limit. As mentioned, the k → 0 limit should also be related to positivity issues. The proper positivity requirement replacing Criterion (FRGC2) yet has to be found, however some constraint will certainly be needed. Concerning Criterion (FRGC1) the premise in the if then statement preceeding Equation (2.12View Equation) has to be justified by external means or taken as a working hypothesis. In principle one can also adopt the viewpoint that the quantum gravity counterpart of Equation (2.12View Equation) discussed in Section 4 simply defines the effective action for quantum gravity whenever a solution meets Criterion (FRGC1). The main drawback with this proposal is that it makes it difficult to include information concerning Criterion (FRGC2). However difficult and roundabout a functional integral construction is, it allows one to incorporate ‘other’ desirable features of the system in a relatively transparent way.

We shall therefore also in the application to quantum gravity assume that a solution Γ k of the cutoff independent FRGE (2.12View Equation) satisfying Criterion (FRGC1) comes from an underlying functional integral. This amounts to the assumption that the renormalization problem for Γ k,Λ defined in terms of a functional integral can be solved and that the limit lim Λ→ ∞ Γ k,Λ can be identified with Γ k. This is of course a rather strong hypothesis, however its self-consistency can be tested within the FRG framework.

To this end one truncates the space of candidate continuum functionals trunc Γ k [φ] to one where the initial value problem for the flow equation (2.12View Equation) can be solved in reasonably closed form. One can then by ‘direct inspection’ determine the initial data for which a global solution exists. Convexity of the truncated lim Γ trunc[φ] k→0 k can serve as guideline to identify good truncations. If the set of these initial data forms a nontrivial unstable manifold of the fixed point trunc trunc S∗ [φ] = limk → ∞ Γk [φ], application of the above criterion suggests that trunc Γ k can approximately be identified with the projection of the continuum limit (lim Λ→ ∞ Γ Λ,k)trunc of some Γ Λ,k computed by other means. The identification can only be an approximate one because in the Γ tkrunc evolution one first truncates and then evolves in k, while in (lim Γ [φ])trunc Λ→ ∞ Λ,k one first evolves in k and then truncates. Alternatively one can imagine to have replaced the original dynamics by some ‘hierarchical’ (for want of a better term) approximation implicitly defined by the property that (lim Λ→ ∞ Γ hΛi,ekr[φ])trunc = lim Λ→∞ Γ hΛie,kr[φ ] (see [75] for the relation between a hierarchical dynamics and the local potential approximation). The existence of an UV fixed point with a nontrivial unstable manifold for Γ trunc k can then be taken as witnessing the renormalizability of the ‘hierarchical’ dynamics.


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