Our first goal thus is to construct the infinite cut-off limit of the background effective action in the covariant background field formalism to all orders of the loop expansion. It turns out that this can be done only if infinitely many essential couplings are allowed, so even the trunctated functional integral based on Equation (3.16) is not renormalizable in the strict sense. However, once one allows for infinitely couplings strict cutoff independence () can be achieved. Remarkably, for Equation (3.1) the generalized beta function for a generating functional of these couplings can be found in closed form (see Equation (3.88) below). This allows one to study their RG flow in detail and to prove the existence of a non-Gaussian UV stable fixed point. One also finds a Gaussian fixed point which is not UV stable.
The main principle guiding the renormalization are generalized Ward identities for the Noether currents (3.26) and the conformal currents (3.27). The general solution of these generalized Ward identities suggests a space of actions which is stable under the renormalization flow. One finds that by suitable redefinitions the general solution can always be brought into the form
As described above in the sigma-model perturbation theory we use dimensional regularization and minimal subtraction. Counter terms will then have poles in rather than containing positive powers of the cutoff . The role of the scale is played by the renormalization scale , the fields and the couplings at the cutoff scale are called the “bare” fields and the “bare” couplings, while the fields and couplings at scale are referred to as “renormalized”. The fact that the ansatz (3.56) ‘works’ is expressed in the following result:
Result (Generalized renormalizability) [154, 155]:
To all orders in the sigma-model loop expansion there exist nonlinear field renormalizations such that for any prescribed bare generating coupling functional there exists a renormalized such that
A subscript ‘’ denotes the bare fields while the plain symbols refer to the renormalized ones and
similarly for . Notably no higher order derivative terms are enforced by the renormalization process;
strict cutoff independence can be achieved without them. However the fact that and differ
marks the deviation from conventional renormalizability. The pre-factor also has a physical
interpretation: To lowest order it is for pure gravity the conformal factor in a Weyl transformation
of a generic four-dimensional metric with two Killing vectors in adapted
The derivation of this result is based on a reformulation of the class of QFTs based on Equation (3.56) as a Riemannian sigma-model in the sense of Friedan . This is a class of two-dimensional QFTs which is also (perturbatively) renormalizable only in a generalized sense, namely by allowing for infinitely many relevant couplings. The generating functional for these couplings in this case is a (pseudo-)Riemannian metric on a “target manifold ” of arbitrary dimension and field coordinates , where is the two-dimensional “base manifold”. The renormalization theory of these systems is well understood. A brief summary of the results relevant here is given in Appendix B.3.
The systems (3.56) can be interpreted as Riemannian sigma models where the target manifold of a special class of “warped products” (see Equation (3.59) below) and the fields are . The relation between the quantum theory of these Riemannian sigma-models and the QFT based on Equation (3.56) will roughly be that one performs an infinite reduction of couplings in a sense similar to [236, 160, 174]. The generating functional is parameterized by functions of variables, while the generating functional in Equation (3.56) amounts to one function of one variable. Thus “” many couplings are reduced to “” many couplings. As always in a reduction of couplings the nontrivial point is that this reduction can be done in a way compatible with the RG dynamics. The original construction in [236, 160] was in the context of strictly renormalizable QFTs with a finite number of relevant couplings. In a QFT with infinitely many relevant couplings (QCD in a lightfront formulation) the reduction principle was used by Perry–Wilson . A general study of an ‘infinite reduction’ of couplings has been performed in .
The reduction technique used here is different, but essentially Equation (3.68) below plays the role of the reduction equation. Apart from the different derivation and the fact that the reduction is performed on the level of generating functionals, the main difference to a usual reduction is that Equation (3.68) also involves nonlinear field redefinitions without which the reduction could not be achieved here. The reduction equation (3.68) thus mixes field redefinitions and couplings. From the viewpoint of Riemannian sigma-models this amounts to the use of metric dependent diffeomorphisms on the target manifold, a concept neither needed nor used in the context of Riemannian sigma-models otherwise.
Since Riemannian sigma-models have been widely used in the context of “strings in curved spacetimes” it may be worthwhile to point out the differences to their use here:
The situation changes drastically if one considers Riemannian sigma-models where the target manifold is one the warped products (3.59) below. The Problem 5 is absent on the basis of the following Non-renormalization Lemma, the Problem 6 is evaded because the Ricci-type flow arising at first order is constant  while to higher orders the asymptotic safety property to be described strikes:
Non-renormalization Lemma :
The field is nonlinearly renormalized but once it is renormalized arbitrary powers thereof (defined by multiplication pointwise on the base manifold) are automatically finite, without the need of additional counterterms. In terms of the normal product defined in Appendix B.3. for an arbitary (analytic) function .
Needless to say that the same is not true for or any other of the quantum fields . As a consequence of this Non-renormalization Lemma the renormalization flow equations for the generating functional (the counterpart of ) can be consistently interpreted as an equation for a classical field, which we also denote by since the quantum field can be manipulated as if it was a classical field. The resulting flow equations then take the form of a recursive system of nonlinear partial integro-differential equations, which are studied in Section 3.4.
We now describe the derivation of these results in outline; the full details can be found in [154, 155]. The class of warped product target manifolds relevant for Equation (3.56) is of the form
Here we combined the field vector (3.9) with to a -dimensional vector , and the metric (3.60) refer to this coordinate system. Further are real parameters kept mainly to illustrate that they drop out in the quantities of interest. The metric is chosen such that for the parameter values , the Lagrangian (3.56) can be written in the form that they have the following structure:
The are constants defined through the curvature scalars of . The are differential polynomials in invariant under constant rescalings of and normalized to vanish for constant . The first three areboth sides of Equation (3.66) the argument is the renormalized field. The renormalized function is allowed to depend on ; specifically we assume it to have the form
Combining Equations (3.60, 3.65, 3.66) and (3.63) one finds that the first order poles cancel in the renormalized Lagrangian iff the following “reduction condition” holds:
For the derivation of Equations (3.68, 3.70) we fixed a coordinate system in which the target space metric takes the form (3.60). Under a change of parameterization the reduction condition (3.68) should transform covariantly, and indeed it does. The constituents transform as[110, 41] and is one of the main advantages of the covariant background field expansion. The relations (3.73) can be used to convert the solutions (3.70) of the finiteness condition into any desired coordinate system on the target space. The coordinates and used in Equation (3.60) are adapted to the Killing vector and the conformal Killing vectors , .
This completes the renormalization of the Lagrangian . The nonlinear field redefinitions alluded to in Equation (3.57) are explicitly given by Equation (3.70). The function plays the role of a generating function of an infinite set of essential couplings. In principle it could be expanded with respect to a basis of -independent functions of with -dependent coefficients, the couplings. Technically the fact that these couplings are essential (in sense defined in the introduction) follows from Equations (3.27). Since is a nontrivial function on the base manifold, the Lagrangian is a total divergence on shell if and only if , or . The first case corresponds to the classical Lagrangian (3.16), the second case was studied (in a different context) by Tseytlin . In the case the identity reflects the fact that the overall scale of the metric is an inessential parameter (see Appendix A). The renormalization flow associated with the coupling functional will be studied in the next Section 3.4.
The fields themselves, here to be viewed as a collection of inessential parameters, are likewise subject to flow equations. Recall from Equation (3.65) the relation between the bare and the renormalized fields, where , while , have been computed in Equation (3.70) and depend on . Since the bare fields are -independent, the renormalized fields have to carry an implicit -dependence through . (This is analogous to the situation in an ordinary multiplicatively renormalizable quantum field theory, where the coupling dependence of the wave function renormalization induces a compensating -dependence of the renormalized fields governed by the anomalous dimension function.) The flow equations involve functional derivatives with respect to the field. For any functional of we set
We proceed with the renormalization of composite operators. Again we borrow techniques from Riemannian sigma models (see Appendix B.3). The normal product of scalar, vector, and tensor operators on the target manifold is defined in Equation (B.70). For generic composite operators of course the bare operator viewed as a function of the bare couplings and of the bare fields will have a different functional form from the renormalized one viewed as a function of the renormalized couplings and fields. An important exception was already described in the above ‘Non-renormalization Lemma’ for functions of only. This is specific to the system here. Another class of operators for which similar non-renormalization results hold are conserved Noether currents; this is a feature true in general. In the case at hand the relevant Noether currents are Equation (3.26), the current , and the “would-be” energy momentum tensor . In terms of the normal product (B.70) the corresponding non-renormalization results read both sides is the renormalized field. Starting from the fact that the right-hand-side of Equation (3.80) is -independent one can derive an non-autonomous flow equation 
© Max Planck Society and the author(s)