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 [1462116622929]. Here we shall mainly use the effective average action 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 [157].

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 , for other types of FRGEs the discussion is similar though. The effective average action interpolates between the bare action and the above, initially regulated, effective action , according to

Roughly speaking one should think of as the conventional effective action but with only the momentum modes in the range integrated out. The limit in Equation (2.7) will in fact differ from by a -loop determinant (see Appendix C.2). For the following discussion the difference is inessential and for (notational) simplicity we will identify with . Equation (2.7) also presupposes that for fixed UV cutoff the limit 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.8) below). Its counterpart for reads

where the functional measure includes mostly momentum modes in the range . This can be done by multiplying the kinematical measure by a suitable mode suppression factor
with a suitable quadratic form . From Equation (2.8) one can also directly verify the alternative characterization (1.12).

The precise form of the mode suppression is inessential. In the following we outline a variant which is technically convenient. Here is a quadratic form in the fields defined in terms of a kernel chosen such that both and define integral operators of trace-class on the function space considered. We write for the integral operator and for its trace. The other properties of the kernel are best described in Fourier space, where acts as , with , the Fourier transform of and similarly for the kernel (where we omit the hat for notational simplicity). The UV cutoff renders Euclidean momentum space compact and Mercer’s theorem then provides simple sufficient conditions for an integral operator to be trace-class [157]. We thus take the kernel to be smooth, symmetric in , and such that

for all continuous functions (similarly for ). 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 and (with foresight) . We refer to Appendix C for more details.

The presence of the extra scale allows one to convert Equation (2.8) into a functional differential equation [22822929],

known as the functional renormalization group equation (FRGE) for the effective average action. For convenience we include a quick derivation of Equation (2.11) in Appendix C. In the second line of Equation (2.11) we spelled out the trace using that is trace-class. Further is the integral operator whose kernel is the Hessian of the effective average action, i.e. , and is the integral operator in Equation (2.8).

For finite cutoffs the trace of the right-hand-side of Equation (2.11) 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 (2.11) is finite as long as the inverse of defines a bounded operator. For finite UV cutoff one sees from the momentum space version of Equation (B.2) in Section 3.4 that this will normally be the case. The trace-class property of the mode cutoff operator (for which Equation (2.10) is a sufficient condition) also ensures that the trace in Equation (2.11) can be evaluated in any basis, the momentum space variant displayed in the second line is just one convenient choice.

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

All this has been for a fixed UV cutoff . 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 dependence in the trace on the right-hand-side of Equation (2.11), and second removal of the UV cutoff in 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 defines a bounded operator. If now is independently known to have a finite and nontrivial limit as , the explicit dependence carried by the term is harmless and the trace always exists. Roughly this is because the derivative kernel has support mostly on a thin shell around , so that the (potentially problematic) large behavior of the other factor is irrelevant (cf. Appendix C.2).

The second aspect of course relates to the traditional UV renormalization problem. Since came from a regularized functional integral it will develop the usual UV divergencies as one attempts to send to infinity. The remedy is to carefully adjust the bare action – that is, the initial condition for the FRGE (2.11) – in such a way that functional integral – viz. the solution of the FRGE – is asymptotically independent of . Concretely this could be done by fine-tuning the way how the parameters in the expansion depends on . However the FRGE method in itself provides no means to find the proper initial functional . Identification of the fine-tuned lies at the core of the UV renormalization problem, irrespective of whether is defined directly via the functional integral or via the FRGE. Beyond perturbation theory the only known techniques to identify the proper start directly from the functional integral and are ‘constructive’ in spirit (see [19548]). Unfortunately four-dimensional quantum field theories of interest are still beyond constructive control.

One may also ask whether perhaps the cutoff-dependent FRGE (2.11) itself can be used to show that a limit 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 [200123]). 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 , , such that the solution of the FRGE (2.11) remains finite as , then the limit has to obey the cut-off independent FRGE

Conversely, under the above premise, this equation should have at least one solution with a finite limit . This limit can now be identified with the renormalized fixed point action . It is renormalized because by construction the cutoff dependencies have been eaten up by the ones produced by the trace in Equation (2.11). It can be identified with a fixed point action because lowering amounts to coarse graining, and 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 , unitarity can in principle be tested retroactively, and is expected to hold only in the limit . Unfortunately this is a very indirect and retroactive criterion. One of the roles of the bare action is to encode properties which are likely to ensure the desired properties of . In theories with massless degrees of freedom the 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 limit is essential to probe stability/positivity/unitarity.

One aspect of positivity is the convexity of the effective action. The functional equations (2.11, 2.12) do in itself “not know” that 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 has positive spectral values for all (cf. Equation (C.14)). If no blow-up occurs in the flow the limit 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 must be local, i.e. a differential polynomial of finite order in the fields so as to end up with an effective action 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 (2.12) which exists globally in (for all ) can reasonably be identified with the continuum limit of the effective average action constructed by other means. For such a solution is the full quantum effective action and is the fixed point action.

(FRGC2) For a unitary relativistic quantum field theory positivity/unitarity must be tested retroactively from the functional derivatives of .