The goal will be to construct the renormalized effective action based on a geodesic background-fluctuation split and as discussed in Section 3.2.2. The variant where the normal coordinate field is the dynamical variable is used as well as dimensional regularization. The latter has the advantage that the additional curvature dependent terms in the measure in Equation (B.28) do not contribute. The explicit counterterms have been computed in minimal subtraction and all scheme dependent quantities will refer to this scheme. The dynamical scalar fields will be denoted by , the background fields by and the normal coordinate fields by .

For the purposes of renormalization it is useful to consider an extended Lagrangian of the form

Here , is a collection of generalized couplings/sources (of the tensor type indicated by the index structure) that depend both on the fields and explicitly on the point in the “base space”. The latter is a two-dimensional Riemannian space with a classical background metric , extended to dimensions in the sense of dimensional regularization, and . The action functional is . The explicit -dependence of the sources allows one to define local composite operators via functional differentiation after renormalization. In addition the scalar source provides an elegant way to compute the nonlinear renormalizations of the quantum fields in the background expansion [110].Recall that the geodesic background-fluctuation split involves decomposing the fields into a background field configuration and a power series in the fluctuation fields whose coefficients are functions of . The series is defined in terms of the unique geodesic from the point (with , say) in the target manifold to the (nearby) point (with ), where is the tangent to vector at . We shall write for this series, and refer to , , and as the full field, the background field, and the quantum field, respectively. On the bare level one starts with fields which upon renormalization are replaced by . The expansion of the Lagrangian (B.57) can be computed from

The derivatives can be reduced to background covariant derivatives , where are the Christoffel symbols of evaluated at . One easily verifies that along with and also transform as vectors under background field transformations. Hence all monomials built from a covariant tensor in , by contracting it with combinations of , , , will be invariant under background field transformations. These are precisely the monomials entering the expansion of a reparameterization invariant Lagrangian like . For example for the metric part one finds from Equation (B.58) etc., with of order in . Here is the Riemann tensor of evaluated at .Both and transform as vectors under reparameterizations of the background fields . The -th order term in Equation (B.58) is of order in and at most quadratic in . The generalized couplings/sources are evaluated at and transform according to their respective tensor type under . In total this renders each term in Equation (B.58) individually invariant under these background field transformations. The term quadratic in is used to define the propagators, all other terms are treated as interactions. Due to the expansion around a nontrivial background field configuration does not have a standard kinetic term, e.g. the first term in Equation (B.57) gises rise to To get standard kinetic terms one introduces the components , with respect to a frame field satisfying . Rewriting in terms of the frame fields a potential is generated which combines with the other terms in , but the have a standard kinetic term and are formally massless. These fields are given a mass which sets the renormalization scale. The development of perturbation theory then by-and-large follows the familar lines, the main complication comes from the complexity of the interaction Lagrangians . In addition nonlinear field renormalizations are required; the transition function can be computed from the differential operator in Equation (B.57) below. For our purposes we in addition have to allow for a renormalization of the background fields. As usual we adopt the convention that the fields remain dimensionless for base space dimension .

With this setting the bare couplings/sources have dimension and are expressed as a dimensionless sum of the renormalized and covariant counter tensors built from . A suitable parameterization is

Here denotes differentiation with respect to at fixed . The quantities and contain poles in (but no other type of singularities) whose coefficients are defined by minimal subtraction. Except for they depend on only; in addition depends quadratically on and , but the quadratic forms with which they are contracted again only depend on . All purely -dependent counter tensors are algebraic functions of , its covariant derivatives, and its curvature tensors. and specifically are linear differential operators acting on scalars and vectors on the target manifold, respectively. The combined pole and loop expansion takes the form for any of the quantities . The residue of the simple pole is denoted by . We do not include explicitly powers of the loop counting parameter in Equation (B.62). For the purely -dependent counter terms of interest here they are easily restored by inserting and utilizing the scaling properties listed below. However once is ‘deformed’ into a nontrivial function of the ‘scaling decomposition’ (B.62) no longer coincides with the expansion in powers of and the former is the fundamental one. Under a constant rescaling of the metric the purely -dependent counter term coefficients transform homogeneously as follows In principle the higher order pole terms , , are determined recursively by the residues of the first order poles via “generalized pole equations”. The latter can be worked out in analogy to the quantum field theoretical case (see [57, 162]). Taking the consistency of the cancellations for granted one can focus on the residues of the first order poles, which we shall do throughout. Explicit results for them are typically available up to and including two loops [84, 220, 57, 110, 162].
For the metric and the dilaton beta functions also the three-loop results are
known^{1}:

Some explanatory comments should be added. First, in addition to the minimal subtraction scheme the above form of the counter tensors refers to the background field expansion in terms of Riemannian normal coordinates. If a different covariant expansion is used the counter tensors change. Likewise the standard form of the higher pole equations [57, 162] is only valid in a preferred scheme. For instance for the metric counter terms in this scheme additive contributions to of the form are absent [110]. Note that adding such a term for leaves the metric beta function in Equation (B.76) below unaffected, provided is functionally independent of .

So far only the full fields entered, on the bare and on the renormalized level. Their split into background and quantum contributions is however likewise subject to renormalization. A convenient way to determine the transition function from the bare to the renormalized quantum fields was found by Howe, Papadopolous, and Stelle [110]. In effect one considers the inversion of the normal coordinate expansion of the renormalized fields. If in Equation (B.61, B.66) is regarded as a differential operator acting on the second argument of this function, i.e. on ,

one obtains the desired relation by inversion. To lowest order yields At each loop order the coefficient is a power series in whose coefficients are covariant expressions built from the metric at the background point.With all these renormalizations performed the result can be summarized in the proposition [110, 162] that the source-extended background functional

defines a finite perturbative measure to all orders of the loop expansion. The additional source here is constrained by the requirement that . The key properties of are:- It is invariant under reparameterizations of the background fields .
- It obeys a simple renormalization group equation (which would not be true without the F-source).
- A generalized action principle holds that allows one to construct local composite operators of dimension , by variation with respect to the renormalized sources.

Let , , be a scalar, a vector, and a symmetric tensor on the target manifold, respectively. ‘Pull-back’ composite operators of dimension 0,1,2 are defined by [162]

The functional derivatives here act on functionals on the target manifold at fixed , e.g. . For in addition the dependence of the counter terms on has to be taken into account, so that . Further is the bare Lagrangian regarded as a function of the renormalized quantities. The contractions on the base space are with respect to the background metric . The additional total divergence in the last relation of Equation (B.70) reflects the effect of operator mixing. The normal products as given in Equation (B.70) still refer to the functional measure as defined by the source-extended Lagrangian. After all differentiations have been performed the sources should be set to zero or rendered -independent again to get the composite operators e.g. for the purely metric sigma-model.The definition (B.70) of the normal products is consistent with redefinitions of the couplings/sources that change the Lagrangian only by a total divergence. The operative identities are

for a scalar . They entail Moreover the invariance of the regularization under reparameterizations of the target manifold allows one to convert the reparameterization invariance of the basic Lagrangian (B.57) into a “diffeomorphism Ward identity” [201, 162]: with . The Lie derivative terms on the right-hand-side are the response of the couplings/sources under an infinitesimal diffeomorphism . Thus may be viewed as a “diffeomorphism current”. The last term on the right-hand-side is the (by itself finite) “equations of motion operator”. In deriving Equation (B.73) the following useful consistency conditions ariseSo far the renormalization was done at a fixed normalization scale . The scale dependence of the renormalized couplings/sources is governed by a set of renormalization functions which follow from Equation (B.61). For a counter tensor of the form (B.62) it is convenient to introduce

which in view of Equation (B.62) can be regarded as a parametric derivative of . ThenThe associated renormalization group operator is

For example the dimension 0 composite operators in Equation (B.70) obey and similar equations hold for the dimension 1, 2 composite operators.An important application of this framework is the determination of the Weyl anomaly as an ultraviolet finite composite operator. We shall only need the version without vector and scalar functionals. The result then reads [201, 220, 162]

Here the so-called Weyl anomaly coefficients enter: where and are the renormalization group functions of Equation (B.76) and These expressions hold in dimensional regularization, minimal subtraction, and the backgound field expansion in terms of normal coordinates. Terms proportional to the equations of motion operator have been omitted. The normal-products (B.70) are normalized such that the expectation value of an operator contains as its leading term the value of the corresponding functional on the background, , where the subleading terms are in general nonlocal and depend on the scale . For the expectation value of the trace anomaly this produces a rather cumbersome expression (see e.g. [220]). As stressed in [201] the result (B.70), in contrast, allows one to use as a simple criterion to select functionals which ‘minimize’ the conformal anomaly.The Weyl anomaly coefficients (and the anomaly itself) can be shown to be invariant under field redefinitions of the form

with functionally independent of the metric. Roughly speaking Equation (B.82) changes the beta function by a Lie derivative term that is compensated by a contribution of the diffeomorphism current to the anomaly which amounts to [201]. It is important to distinguish these diffeomorphisms from field renormalizations like Equation (3.65, 3.70) that depend on the metric. Although formally Equation (B.82) amounts to in Equation (3.65); clearly one cannot cancel one against the other. The distinction is also highlighted by considering the change in the metric counter terms under Equation (B.82). Without further specifications this would not be legitimate for a -dependent vector. Although the Lie derivative term in Equation (B.83) drops out when recomputing directly as a parametric derivative, in combinations like the term in the metric beta function of Equation (B.76) induces an effective shift Similarly is shifted to and the Weyl anomaly coefficients are invariant.In the context of Riemannian sigma-models is usually interpreted as a “string dilaton” for the systems (B.57) defined on a curved base space. If one is interested in the renormalization of Equation (B.57) on a flat base space, on the other hand plays the role of a potential for the improvement term of the energy momentum tensor. This role of can be made manifest by rewriting Equation (B.79) by means of the diffeomorphism Ward identity. Returning to a flat base space one finds [201, 162]

where again terms proportional to the equations of motion operator have been omitted. Here is the ‘naive’ improvement term while the additional total divergence is induced by operator mixing.The functions and are linked by an important consistency condition, the Curci–Paffuti relation [57]. We present it in two alternative versions,

The first version displays the fact that the identity relates various -dependent counter terms without entering. In the second version is introduced in a way that yields an identity among the Weyl anomaly coefficients. It has the well-known consequence that is constant when vanishes: where is the central charge of energy momentum tensor derived from Equation (B.57).

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