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.
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
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[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 known1: and . For the results are [162, 57, 220] [162, 110, 220]  but are not needed here.
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 . 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 . 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 ,
With all these renormalizations performed the result can be summarized in the proposition [110, 162] that the source-extended background functional
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 
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[201, 162]:
So 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
The associated renormalization group operator is
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]). As stressed in  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 formindependent 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 . 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
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]
The functions and are linked by an important consistency condition, the Curci–Paffuti relation . We present it in two alternative versions,
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