Before turning to the formalism let us briefly comment on the status of the background field configuration. First one should stress that the term “background field” in this context does not refer to a solution of the classical field equations. It is an auxiliary device used to formulate covariance properties that constrain the renormalization flow. The dependence on the background configuration is controlled by splitting Ward identities that, roughly, ensure that nothing depends on the way how the integration variable in the functional integral is decomposed into a reference configuration and a fluctuation field. Once a source free condition is imposed the “background” gets related to the expectation value of the fluctuation field through a consistency condition involving the full quantum effective action (see Equations (2.51) and (B.53) below).

When applied to quantum gravity, the background field formalism can be viewed as giving rise to a formulation with a state-dependent dynamically adjusted reference metric. The bare manifold is initially equipped with a reference geometry, but rather than being external it eventually gets related to the average of the quantum metric in a self-consistent manner. As an incomplete classical analogy one may take the variational principles for general relativity which in addition to the dynamical metric invoke a fiducial background metric (see e.g. [74] and references therein). The latter only provides the desired covariance properties and in the presence of generic boundary conditions serves to define conserved quantities relative to the background.

As a further clarification one should add that in the case of gauge theories the on-shell projections of the Green’s functions (e.g. S-matrix elements) computed from the background effective action are not assumed to be physical quantities from the outset. In particular in the case of gravity the precise role of the on-shell projections remains to be understood. The main reason for focusing on the background effective action here are its ‘benign’ properties under UV renormalization. The background covariance constrains the renormalization flow, both on the level of a Wilsonian action and on the the level of an effective average action.

The background field formalism comes in two main variants: one based on a linear background-fluctuation split and the other based on a geodesic background-fluctuation split. The latter is used to cope with field reparameterization symmetries. Both variants can be applied to non-gauge theories (regular Lagrangians) and gauge theories (singular Lagrangians). We briefly discuss these variants consecutively, later on the variants in Sections B.2.2 and B.2.3 will be used.

In this simple case the use of a background field formalism is ‘overkill’. However it is a convenient way to introduce the relevant structures. The starting point is again the generating functional for the connected Green’s functions, but with a modified coupling to the source . Specifically, one linearly decomposes the quantum field, i.e. the integration variable , according to , where is a background configuration to be adjusted later and is the fluctuation field, i.e. the new integration variable. In the corresponding generating functional only not the complete field is coupled to the source. For the sake of illustration we allow the action to depend explicitly on the background field , i.e. . One introduces

and for a given its Legendre transform Assuming for simplicity that is differentiable everywhere, for an extremizing configuration one has i.e. can be interpreted as the normalized expectation value of the fluctuation field. The counterpart of Equation (B.8) is Differentiating with respect to gives We shall refer to identities of this form as “splitting Ward identities”. Here Equation (B.17) expresses the fact that that the “linear splitting” symmetry , is violated only by the explicit background dependence of the action.The condition that the source in Equation (B.15) vanishes is usually solved by formal power series inversion and then gives as the only solution, . More generally the vanishing of defines locally as a function of , say, . For such configurations becomes a functional of a single field

which obeysWe briefly mention two applications of the relations (B.65), both for actions without explicit background dependence. The relation (B.17) is often used for constant background field configurations . It then implies that vertex functions with vanishing external momenta can be written as derivatives of vertex functions with a smaller number of legs

In the context of renormalization theory the following discussion (taken from Howe et al. [110]) illustrates the point of the identity (B.17). Expanding an interaction monomial like one gets where in principle all interaction vertices could be renormalized differently. However, because of Equation (B.17) this does not happen, the wave function renormalization constants for and are actually the same: This entails that the counterterms are functionals of the full field , not of and separately. Relations like Equations (B.17) – referred to as “linear splitting Ward identities” in [110] – thus provide a crucial simplification in the descrition of the renormalization flow, once the background-fluctuation split has been adopted for other reasons.Good reasons to adopt such a split exist in theories with symmetries, which can be local gauge symmetries, or field reparameterization symmetries, or both. In all situations the background field method offers key advantages in that it can produce an effective action which is an invariant functional of its argument. Via the above splitting principle this then greatly restricts the form of the (Wilsonian) renormalized action. In a non-background field formalism these symmetries in contrast have to be imposed by relating possibly noninvariant terms or pieces of the renormalized action via conventional Ward identities, like Equation (A.9) in the case of field reparameterizations.

We first describe the background field technique for a non-gauge theory where reparameterization invariance in field space is aimed at, and then for a diffeomorphism gauge theory (the case of Yang–Mills theories runs completely parallel). Finally we mention the setting where both gauge and field reparameterization invariance is aimed at. General references are [68, 49, 223, 224, 129, 178].

Here reparameterization invariance in field space is aimed at; the original construction is due to Honerkamp et al. [107, 109]. Since invariance under local field redefinitions is a hallmark of physical quantities this field reparameterization invariant effective action is an object much more intrinsic to the field theory under consideration. Morally speaking in this technique the field reparameterization Ward identity (A.9) is built in, and does not have be imposed along with the solution/definition of the functional integral. For the construction of the covariant effective action the field configuration space is equipped with a metric and the associated metric connection. The Lagrangian is assumed to be reparameterization in variant in the sense that

Here is a vector field on the configuration space and is the action associated with . The second equation is a consequence of the first; the quantity can be viewed as a “diffeomorphism current”. For example , if depends algebraically on and , and with obvious generalizations to higher derivative theories. Formally such a reparameterization invariance can always be achieved. Naturally it arises for a scalar field theory when the metric in field space enters via the kinetic term in the Lagrangian where the other terms are supposed to preserve Equations (B.23, B.24) The prototype example of such systems are Riemannian sigma-models, for which the resulting “covariant background field method” is summarized in Appendix B.3. From a Wilsonian perspective it is natural to allow for higher derivative interactions and to use only the invariance (B.23, B.24) for the formulation of the framework.One now describes the configurations in terms of an arbitrary (off-shell) background configuration and geodesic normal coordinates , which are new dynamical fields. That is, a nonlinear background-fluctuation split is used. Here is the function on such that its value gives the endpoint of the (locally unique) geodesic in connecting to and having as the tangent vector at . The normal coordinate expansion of is a power series in with coefficients built from the Christoffel symbols of evaluated at . We shall also need the inverse series , defined by . To quadratic order one has

One sees that transforms like a vector with respect to local reparameterizations of the background field and like a scalar with respect to reparameterizations of . Later on we shall need to relate functional derivatives with respect to the different fields. The relevant relations are One can also verify that for a function an -th ordinary derivative with respect to ’s becomess a symmetrized covariant derivative with respect to ’s with viewed as a function of . Finally the invariant measures are related by With these geometrical properties at hand it is now straightforward to construct generating functionals which are reparameterization invariant functionals of their arguments. Starting with in Equation (B.13) (with being not explicitly background dependent) one simply replaces the linear but noncovariant source coupling by the covariant but nonlinear one , where is a co-vector with respect to background field reparameterizations. This gives The first expression shows that the dependence on the background configuration enters only through the source term; in the second expression we changed variables according to . Correspondingly there are two ways to introduce a mean fluctuation field which treat and as the dynamical field, respectively, with mean fields and . Both are functions of , and is implicitly defined by Equation (B.30). The Legendre transforms are respectively, which gives for the extremizing configurations The Legendre transforms satisfy the functional integro-differential equations with the arguments of the source terms as in Equation (B.32). The reason for carrying both variants along is to highlight that the effective action based on a geodesic split is at least formally a coordinate independent concept, where patches in field space around a reference configuration are invariantly described. Technically the use of the normal coordinate field as a dynamical variable is simpler but presumably not indispensible. The -point functions are related by [50] where the left-hand-side is the symmetrized covariant derivative with respect to (as in in Equation (B.27) for ) and the right-hand-side is the vertex (“1-particle irreducible”) function of the normal coordinate fields.For later reference let us also note that once in the source-free condition is imposed, with as the only solution within formal power series inversion, one has

which is the counterpart of Equation (B.17). Comparing with Equation (B.29), one sees that the functional integral for reads where the source is constrained by the requirement that (given by the same functional integral with a insertion) vanishes. For generalized Riemannian sigma-models the perturbative renormalization of this generating functional is summarized in Appendix B.3.We now describe two types of Ward identities for these systems: diffeomorphism type Ward identities and the nonlinear splitting Ward identities mentioned earlier. The former are really generalized Ward identites relating different theories in the sense that a compensating change in the metric tensor is needed. On the classical level an example is Equation (B.24); only if admits Killing vectors and one takes for one of the Killing vectors does Equation (B.24) reduce to a conservation equation proper, with being the associated Noether current. In fact, Equation (B.24) can be promoted to a “Diffeomorphism Ward identity” in the quantum theory, at least perturbatively [201, 162] and presumably also in a non-perturbative formulation. In perturbation theory the “equations of motion operator” appearing on the right hand side of Equation (B.24) is a finite operator, i.e. it is the same when viewed as a functional of the bare fields and couplings, and when viewed as a function of the renormalized fields and couplings. Once the second term on the left-hand-side has been defined in terms of normal products, the diffeomorphism current must be finite as well.

For the background effective action a similar diffeomorphism Ward identity arises as follows. Since a geodesic is a coordinate independent concept, transforming all of the ingredients in the definition of by an infinitesimal diffeomorphism , , gives

Assuming that the functional integral in Equation (B.29) has been invariantly regularized, and noting that the quantum field in the symmetry variation , , enters linearly, one is lead to the following diffeomorphism Ward identity for [110]: The ‘price’ for the linearity here is that the metric tensor changes as well.This is different in the nonlinear splitting Ward identities, which control the dependence on the background field configuration. If the metric in Equation (B.38) is kept fixed the splitting symmetry , becomes nonlinear (see Equation (B.27)). The corresponding “nonlinear splitting Ward identity” reads as follows:

Here is the average of the matrix field in Equation (B.27) defined as a renormalized composite operator and the variations are at fixed . Conceptually Equation (B.40) expresses the fact that in Equation (B.33) the dependence on the background enters only through the source term. Differentiating this equation with respect to the background at fixed and switching to the normal coordinate field as dynamical variable one finds Equation (B.40), if somewhat formally. A technically cleaner way of arriving at Equation (B.40) is by working with the normal coordinate field throughout, promoting the splitting invariance to a nilpotent operation and defining by variation with respect to a source. For details be refer to Howe et al. [110]. Further nonlinear renormalizations of itself have to be taken into account (see Appendix B.3 for a summary). Note that for a flat geometry on field space the equation reduces to Equation (B.17). In a perturbative construction the nonlinear splitting Ward identity (B.40) has the important consequence that the counterterms depend on the full field only and not on , individually. The use of normal coordinates in Equation (B.40) has technical advantages, in principle however a counterpart for the effective action exists as well [50].

Since we are interested here in diffeomorphism invariant theories we consider this type of gauge invariance. The case of Yang–Mills fields is largely parallel (see [108, 1, 25] for the latter). Let be any diffeomorphism invariant action of a Riemannian metric . Infinitesimally the invariance reads , where

is the Lie derivative of with respect to the vector field . As before we decompose the metric (later the integration variable in the functional integral) into a background and a fluctuation , i.e. . Note that is not assumed to be small in some sense, no expansion in powers of 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 can be decomposed in two different ways,

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 to be constructed will be a functional of the averaged metric , the background metric , and sources, which is invariant under the background field transformations (B.43). We quickly run through the relevant steps.The background generating functional is formally defined by

The measure differs from the naive measure by gauge-fixing and ghost contributions, as well as sources , for the ghosts: Here is the gauge fixing condition, which must be invariant under Equation (B.43), but which we may leave unspecified here. A widely used gauge condition is the background harmonic gauge (see Section 4.1). We shall ignore the problem of global existence of gauge slizes in accordance with the formal nature of the construction. The second term in the exponent is the ghost part; it is obtained along the same as lines in Yang–Mills theory: One applies a genuine gauge transformation (B.42) to and replaces the parameters by the ghost field . The integral over the ghosts and exponentiates the Faddeev–Popov determinant then. It is crucial that the ghost and gauge fixing terms are invariant under the background field transformations (B.43) together with Finally we coupled in Equation (B.45) the ghosts to sources , , for later use. The key fact now is the invariance where is the Lie derivative of the respective tensor type. This invariance property follows from Equation (B.45) if one performs a compensating transformation (B.43, B.46) on the integration variables , , , and uses the invariance of all but the source terms. Importantly, at this point one must assume that the formal measure is diffeomorphism invariant.The background effective action now is defined by

The extremizing source configurations , , are characterized by As usual, if is differentiable everywhere with respect to the sources one can interpret , , as the source dependent expectation values of , , via and Equations (B.49, B.50) are related by formal inversion. Indeed, the extremizing sources are usually constructed by assuming that has a series expansion in powers of the sources with -dependent coefficients; formal inversion of the series then gives a with the property that , and similarly for the ghost sources.It is convenient to regard as a functional of and instead of and . We thus set

The crucial invariance property can then be summarized as where all its arguments transform as tensors of the corresponding rank. Equation (B.52) is a direct consequence of Equation (B.47).Finally we have to switch off the sources. Since has ghost number zero , is always a solution of Equation (B.49). For the metric source this is different. Within the realm of formal power series inversions is always a solution of . Combined with the usual uniqueness assumption it is the only solution, and the “self-consistent background determination” at which the background field method aims at degenerates. Indeed, note that in the dependence drops out, as is prescribed. The expectation value of the full quantum metric, , say, just gives back the prescribed background . This evidently has a somewhat perturbative flavor, although no direct reference to perturbation theory is made.

To go beyond that, we directly impose

as the source-free condition. It adjusts the background used as a reference in the functional integral self-consistently to the dynamical , which is the average of the quantum metric. Equation (B.53) can be viewed as the vanishing of the one-point function for all . The multi-point functions are defined by They should contain the complete information about the system, including the state (see Section 2.3). Their precise physics meaning however remains to be understood. In the case of gauge theories which allow for a perturbative definition of scattering states, the background effective action is known to produce the same perturbative S-matrix as the standard effective action [1, 25].In view of Equation (B.53, B.54) one would like to introduce a functional of a single metric only, whose one-point functions have stationary points. This can be done as follows. We define the final “background effective action” [156] by

For the solutions of Equation (B.53) obtained by formal power series inversion one has and thus recovers the usual definition. Generally Equation (B.55) is invariant under , as desired, and obeys

The background gauge invariant effective action in Section B.2.3 depend parametrically on the choice of the background gauge condition . This dependence is a consequence of the field parameterization dependence of the effective actions based on a linear fluctuation background split. The geometrical approach of Vilkovisky and deWitt [68, 223] is designed to overcome this drawback and at least formally it produces an off-shell effective action with all the desirable properties: It is gauge invariant with respect to the background field, gauge invariant with respect to the mean of the fluctuation field, and independent of the choice of the gauge fixing surface. In brief, the strategy is to project locally onto the gauge invariant subspace and then apply the techniques of Section B.2.2. Since we shall not use this formulation here it may suffice to refer to [129] for a brief survey and to [178, 68, 223] for detailed expositions.

With these remarks we conclude our brief introduction to the background field formalism. In the context of the asymptotic safety scenario the variant from Section B.2.2 has been used in [154, 155] (see Section 3) and the variant from Section B.2.4 in [179, 133, 131] (see Section 4).

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