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

Here is the generating functional for an infinite set of couplings; it defines a function of one real variable whose argument in Equation (3.56) is the field . The fact that the renormalization flow does not force one to leave the (infinite-dimensional) space of actions, Equation (3.56) of course has to be justified by explicit construction. Below we shall show this in renormalized perturbation theory to all loop orders. Since the ansatz (3.56) is based on a symmetry characterization is seems plausible, however, that also a nonperturbatively constructed flow would not force one to leave the space (3.56), though one could certainly start off with a more general ansatz containing for example higher derivative terms.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
coordinates.

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 [84]. 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 [174]. A general study of an ‘infinite reduction’ of couplings has been performed in [11].

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:

- First, the scalar fields in Equation (3.56) parameterize a 4D spacetime metric with 2 Killing vectors (not the position of a string in target space) while the target space metric here (see Equation (3.60) below) has 4 Killing vectors. It is auxiliary and not interpreted as a physical spacetime metric. From the viewpoint of “strings in curved spacetime” the system (3.56) (without matter), on the other hand, describes strings moving on a spacetime with 4 Killing vectors and signature .
- The aim in the renormalization process here is to preserve the conformal geometry in target space, not conformal invariance on the worldsheet (base space) . To achieve this one needs metric dependent diffeomorphisms in target space which, as mentioned before, neither need to be nor have been considered before in the context of Riemannian sigma-models.
- As a consequence of Difference 2 the renormalized fields and become scale dependent and their renormalization flow backreacts on the coupling flow (see Equations (3.75, 3.84) below). This aspect is absent if one naively specializes the renormalization theory of a generic Riemannian sigma-model to a target space geometry which is a warped product (see [221]).
- As will become clear later in the class of warped product sigma-models considered here the Weyl anomaly is overdetermined at the fixed point of the coupling flow. In contrast to a generic Riemannian sigma-model one is therefore not free to adjust the renormalized target space metric such that the Weyl anomaly vanishes and the system is a conformally invariant 2D field theory.
- The renormalization flow in Riemannian sigma-models is of the form , where is the renormalized generating coupling functional (“target space metric”) with the renormalized quantum fields inserted. Conceptually the highly nonlinear but local on the right-hand-side thus is a (very special) composite operator, whose finiteness is guaranteed by the construction (see Section B.3). The fact that this very special composite operator is finite does of course not entail that any other nonlinear composite operator built from or is finite (without introducing additional counterterms). For example or a curvature combination of not occuring in is simply not defined off-hand. This is true no matter how is chosen, so the folklore that one can restrict attention to functionals for which the trace or Weyl anomaly vanishes and get a “finite” QFT is incorrect (see [201] for a discussion). Moreover the Weyl anomaly is itself a (very special) composite operator and the condition for its vanishing is not equivalent to a partial differential equation of the same form for any classical metric. By expanding the quantum fields around a classical background configuration one can convert the condition for a vanishing Weyl anomaly into a condition formulated in terms of a classical metric [220]. However beyond lowest order (that is, beyond the Ricci term) nonlocal terms are generated, and the resulting cumbersome equations are rarely used. As a consequence beyond leading order (beyond Ricci flatness modulo an improvement term) most of the “consistent string backgrounds” (defined by ad-hoc replacing the composite operator by a classical metric in the formula for the Weyl anomaly as a composite operator) are actually not consistent, in the sense that the corresponding metric re-interpreted (ad-hoc) as one with the quantum fields re-inserted does not guarantee the vanishing of the Weyl anomaly in its operator form.
- Even the Ricci flow equations arising at lowest order have the property that for a generic smooth target space metric the flow is often singular towards the ultraviolet [52]. For generic target spaces the Riemannian sigma-models are therefore unlikely to give rise to genuine (not merely effective) quantum field theories.

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 [61] while to higher orders the asymptotic safety property to be described strikes:

Non-renormalization Lemma [154]:

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

where is the metric (3.11) on the symmetric space , is the ‘warp factor’, and is a flat two-dimensional space with Lorentzian metric given by the lower block in the metric If is the scalar curvature of the normalized as in Equation (3.12) the metric (3.60) has scalar curvature so that the warp function parameterizes the inverse curvature radius of the target space.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

In addition to the Killing vectors associated with the metric (3.60) possesses two conformal Killing vectors and , which together with generate the isometries of , i.e. , . Conversely any metric with these conformal isometries can be brought into the above form. Each Killing vector of of course gives rise to a Noether current; the conformal Killing vectors and give rise to currents analogous to those in Equation (3.27). The counterpart of on-shell the relations (3.27) is , . Upon quantization in Equation (3.62) plays the role of the loop counting parameter. In dimensional regularization the -loop counter terms contain poles of order in . We denote the coefficient of the -th order pole by . In principle the higher order pole terms are determined recursively by the residues of the first order poles. Taking the consistency of the cancellations for granted one can focus on the residues of the first oder poles, which we shall do throughout. One can show [154] that they have the following structure: It should be stressed that this is not trivially a consequence of the block-diagonal form of Equation (3.60), rather the properties (3.12) enter in an essential way.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 are

The counter terms (3.63) ought to be absorbed by nonlinear field renormalizations, and a renormalization of the function , where is the renormalization scale. Note that on both 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 where the first term ensures standard renormalizability at the 1-loop level – and is determined by this requirement except for the power . The power has no intrinsic significance; one could have chosen a parameterization of the 4D spacetime metric such that the action (3.10) with replaced by was the outcome of the classical reduction procedure. In particular the sectors and are equivalent and we assume throughout.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:

where and is the Lie derivative of , . The -dependence of marks the deviation from conventional renormalizability. Guided by the structure of Equations (3.60) and (3.63) we search for a solution with , where here and later on we also use , for the index labeling. The Lie derivative term with this is The reduction condition (3.68) then is equivalent to a simple system of differential equations whose solution is Here we set and slightly adjusted the notation to stress the functional dependence on . Possibly -dependent integration constants have been absorbed into the lower integration boundaries of the integrals. Throughout these solutions should be read as shorthands for their series expansions in with of the form (3.67). For exampleFor 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

The covariance of the counter terms as a function of the full field is nontrivial [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 [221]. 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

Observe that for any differential or integral polynomial in which is homogeneous of degree , the functional derivative (3.74) just measures the degree, . From Equations (3.65, 3.70) and the -flow (3.83) below one derives where , refer to Equation (3.76) with the solution of Equation (3.83) inserted for . Note that, conceptually, the problems decouple: One first solves the autonomous equation (3.83) to obtain the coupling flow which is then used to specify the right-hand-side of the -flow equation whose solution in turn determines the -flow. The ‘’ derivatives of the solution (3.70) of the reduction condition come out as In we set and anticipated in the notation that this is the conventional beta functions of a coset sigma-model without coupling to gravity. In we absorbed a -dependent additive constant into the lower integration boundary and used , as .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 [154]

For the current the identity follows similarly on general grounds, while the stronger identity is a consequence of the non-renormalization Lemma. The result (3.78) will later turn out to reflect a property of the generalized beta function. For the renormalization of constraints in Equation (3.15) improvement terms are crucial. As in Equation (3.15) we wish to identify the constraints and with the components of the “would be” energy momentum tensor associated with the ‘deformed’ Lagrangian . To this end we decompose the energy momentum tensor for the Lagrangian into a symmetric tracefree part and an improvement term with . The improvement term is trivially conserved but its trace vanishes only on-shell. In contrast to its functional form is not protected by the conservation equation, and for the finiteness of the composite operator the improvement potential has to be renormalized in a way that changes its functional form. This is to say, there is no function such that the bare and the renormalized improvement potential would merely be related by substituting the bare field and the renormalized one , respectively, into . Rather we set and , where is a potentially -dependent constant, is a function of the bare field and , and is a function of the renormalized field and . The finiteness of can then be achieved by relating and (the functions, not their values) according to Here is a differential polynomial in that can be computed from the counterterms in Equation (B.61) of Appendix B.3. Note that as in Equation (3.66) the argument on 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 [155] where is determined by . For a given solution the flow equation (3.81) in principle determines . Finally can be shown to be a finite composite operator. Upon specification of initial data for and satisfying the proper boundary or fall-off conditions in , the composite operator is completely determined. This holds for an arbitrary coupling function . In general the operator will not be trace free. The interpretation of the components of as quantum constraints, on the other hand, requires that the trace vanishes as should be equal to both and . We shall return to this condition below (see Equations (3.100), etc.).http://www.livingreviews.org/lrr-2006-5 |
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