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B.3 Renormalization of Riemannian sigma-models

Here we summarize the results on the UV renormalization of Riemannian sigma-models needed for Section 3. We largely follow the thorough treatment of Osborn [162Jump To The Next Citation Point] and use the results of [84Jump To The Next Citation Point201Jump To The Next Citation Point57Jump To The Next Citation Point110Jump To The Next Citation Point220Jump To The Next Citation Point80Jump To The Next Citation Point].

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 𝒟 ξ measure in Equation (B.28View Equation) 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 i φ, the background fields by ¯i φ and the normal coordinate fields by i ξ.

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

1 μν i j μν i 1 (2) λL (φ; 𝒢) = --¯γ 𝔥ij∂μφ ∂νφ + ¯γ ∂μφ V νi +--¯R (¯γ )D + F. (B.57 ) 2 2
Here 𝒢 = {𝔥,V, D, F}, 𝒢 = 𝒢(φ;x), is a collection of generalized couplings/sources (of the tensor type indicated by the index structure) that depend both on the fields φj and explicitly on the point x in the “base space”. The latter is a two-dimensional Riemannian space with a classical background metric ¯γμν(x ), extended to d dimensions in the sense of dimensional regularization, and ¯(2) (2) R (¯γ ) = R (¯γ )∕ (d − 1 ). The action functional is ∫ d √ -- S [φ;𝒢] = d x ¯γL(φ; 𝒢). The explicit x-dependence of the sources 𝒢 allows one to define local composite operators via functional differentiation after renormalization. In addition the scalar source F provides an elegant way to compute the nonlinear renormalizations of the quantum fields in the background expansion [110Jump To The Next Citation Point].

Recall that the geodesic background-fluctuation split involves decomposing the fields φj into a background field configuration ¯φj and a power series in the fluctuation fields ξj whose coefficients are functions of ¯j φ. The series is defined in terms of the unique geodesic ¯ s ↦→ Σ (φ,ξ;s) from the point ¯ φ (with s = 0, say) in the target manifold to the (nearby) point φ (with s = 1), where ξj is the tangent to Σj vector at φ¯. We shall write φj = Σj (¯φ;ξ,1) for this series, and refer to φ, φ¯, and ξ as the full field, the background field, and the quantum field, respectively. On the bare level one starts with fields j : j ¯ φ B = Σ (φB,ξB;1 ) which upon renormalization are replaced by j j ¯ φ = Σ (φ,ξ;1). The expansion of the Lagrangian (B.57View Equation) can be computed from

∑ n | L(φ;𝒢 ) = 1--d--L(Σ (φ¯, ξ;s);𝒢)|| . (B.58 ) n!dsn s=0 n≥0
The s derivatives can be reduced to background covariant derivatives ¯D μξi = ∂μξi + Γ ijk(φ¯)∂μ¯φiξk, where Γ i (¯φ) jk are the Christoffel symbols of 𝔥 evaluated at ¯φ. One easily verifies that along with ξi and i ∂μ¯χ also ¯ i i i i k Dμξ := ∂μξ + Γ jk(χ¯)∂μ¯χ ξ transform as vectors under background field transformations. Hence all monomials built from a covariant tensor Tij(𝔥 ) in 𝔥,
^ ∂φk-∂φl- ^ Tij(V ∗𝔥)(φ) = ∂φ^i ∂φ^j Tkl(𝔥 )(φ ), φ = V (φ), (B.59 )
by contracting it with combinations of ξi, ∂μ¯φi, D μξi, will be invariant under background field transformations. These are precisely the monomials entering the expansion of a reparameterization invariant Lagrangian like L (φ; 𝒢). For example for the metric part L𝔥(χ) = 1𝔥ij(χ)∂μχi∂ μχj 2 one finds from Equation (B.58View Equation)
¯ ∑ ¯ L 𝔥(φ ) = L𝔥(φ) + n≥1 Ln(φ, ξ), μν i j L1(φ¯, ξ) = 𝔥ij(¯φ)¯γ ∂ μ¯φ ¯D νξ, (B.60 ) [ ] L2(φ¯, ξ) = 12¯γμν 𝔥ij(¯φ)D¯μξi ¯D νξj + Rijkl(¯φ)∂μ¯φi∂ν¯φjξkξl ,
etc., with Ln of order n in ξ. Here Rijkl is the Riemann tensor of 𝔥 evaluated at φ¯.

Both ξi and D¯μξi transform as vectors under reparameterizations of the background fields φ¯↦→ V (¯φ). The n-th order term L n in Equation (B.58View Equation) is of order n in ξ and at most quadratic in ¯ D μξ. The generalized couplings/sources 𝒢 are evaluated at ¯ φ and transform according to their respective tensor type under ¯φ ↦→ V(¯φ ). In total this renders each term in Equation (B.58View Equation) individually invariant under these background field transformations. The term L2 quadratic in ξ is used to define the propagators, all other terms are treated as interactions. Due to the expansion around a nontrivial background field configuration L 2 does not have a standard kinetic term, e.g. the first term in Equation (B.57View Equation) gises rise to 1 μν ¯ ¯ i ¯ j 2¯γ 𝔥ij(φ)D μξD νξ + ... To get standard kinetic terms one introduces the components a a i ξ := Ei (φ¯)ξ, with respect to a frame field satisfying 𝔥ij(¯φ) = Eai (φ¯)Ebj(¯φ)ηab. Rewriting L2 in terms of the frame fields ξa a potential is generated which combines with the other terms in L2, but the ξa have a standard kinetic term and are formally massless. These fields are given a mass μ 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 Ln, n ≥ 3. In addition nonlinear field renormalizations are required; the transition function ξ ↦→ ξB (ξ ) can be computed from the differential operator Z − 1 in Equation (B.57View Equation) below. For our purposes we in addition have to allow for a renormalization ¯ ¯ φB (φ) of the background fields. As usual we adopt the convention that the fields j φB remain dimensionless for base space dimension d ⁄= 2.

With this setting the bare couplings/sources 𝒢B(φB; x) have dimension [μ ]d−2 and are expressed as a dimensionless sum of the renormalized 𝒢(φ;x ) and covariant counter tensors built from 𝒢(φ;x ). A suitable parameterization is

B d−2 𝔥ij = μ [𝔥ij + Tij(𝔥)], [ j -- ] VBμi = μd−2 ZV (𝔥 )iV μj + Nijk(𝔥)∂ μ𝔥jk , (B.61 ) DB = μd−2[Z (𝔥)D + U (𝔥)], FB = μd−2[Z (𝔥)F + Y ].
Here -- ∂μ denotes differentiation with respect to x at fixed φ. The quantities Tij,Nijk,U,Y and ZV − 1,Z − 1 contain poles in (2 − d) (but no other type of singularities) whose coefficients are defined by minimal subtraction. Except for Y they depend on 𝔥ij only; Y in addition depends quadratically on V μi and -- ∂μ𝔥jk, but the quadratic forms with which they are contracted again only depend on 𝔥ij. All purely 𝔥-dependent counter tensors are algebraic functions of 𝔥ij, its covariant derivatives, and its curvature tensors. Z − 1 and ZV − 1 specifically are linear differential operators acting on scalars and vectors on the target manifold, respectively. The combined pole and loop expansion takes the form
∑ ∑ ( )l 𝒪 = ---1---- -1- 𝒪 (ν,l) (B.62 ) (2 − d)ν 2π ν≥1 l≥ν
for any of the quantities Tij,Nijk,U, ZV − 1,Z − 1, Y. The residue of the simple pole is denoted by 𝒪 (1). We do not include explicitly powers of the loop counting parameter λ in Equation (B.62View Equation). For the purely 𝔥-dependent counter terms of interest here they are easily restored by inserting 𝔥∕ λ and utilizing the scaling properties listed below. However once 𝔥 is ‘deformed’ into a nontrivial function of λ the ‘scaling decomposition’ (B.62View Equation) no longer coincides with the expansion in powers of λ and the former is the fundamental one. Under a constant rescaling of the metric the purely 𝔥-dependent counter term coefficients transform homogeneously as follows
𝒪 (ν,l)(Λ −1𝔥) = Λl−1𝒪 (ν,l)(𝔥) for 𝒪 = T , U, ij (ν,l) −1 l (ν,l) V (B.63 ) 𝒪 (Λ 𝔥) = Λ 𝒪 (𝔥) for 𝒪 = Z, Z , N.
In principle the higher order pole terms 𝒪 (ν,l), l ≥ ν ≥ 2, are determined recursively by the residues (1,l) 𝒪 of the first order poles via “generalized pole equations”. The latter can be worked out in analogy to the quantum field theoretical case (see [57Jump To The Next Citation Point162Jump To The Next Citation Point]). 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 [84220Jump To The Next Citation Point57Jump To The Next Citation Point110Jump To The Next Citation Point162Jump To The Next Citation Point]. For the metric 𝔥 and the dilaton D beta functions also the three-loop results are known1:

Ti(1j,1)(𝔥) = Rij, (1,2) 1- klm Tij (𝔥) = 4Riklm Rj , (B.64 ) T (1,3)(𝔥) = 1R kR Rpnmq − 1R Rk Rlmnp − -1-∇ R ∇kR lmn, ij 6 imn jpqk 8 iklj mnp 12 n iklm j
where the three-loop term has been computed independently in [80] and [96]. For D the results are [162Jump To The Next Citation Point57Jump To The Next Citation Point220Jump To The Next Citation Point]
cT 1 U(1,1)(𝔥) = ---, U (1,2)(𝔥) = 0, U(1,3)(𝔥) = --RijklRijkl, (B.65 ) 6 48
where cT is the dimension of the target manifold. For the other quantities one has [162Jump To The Next Citation Point110Jump To The Next Citation Point220Jump To The Next Citation Point]
V j(1,1) 1- 2 j j [Z (𝔥)i] = 2[− ∇ δi + Ri ], [ZV (𝔥)j](1,2) = 1R klj∇ ∇ , i 4 i k l (1,1) 1- 2 Z (𝔥 ) = − 2∇ , (1,2) Z (𝔥 ) = 0, (B.66 ) 3 iklm j Z (𝔥 )(1,3) = − --R R klm ∇i ∇j, 16 1 1 [Nijk(𝔥)](1,1) = -δji∇k − --𝔥jk∇i, 2 4 jk (1,2) 1- jkl [Ni (𝔥)] = 2Ri ∇l.
The expressions for (1,1) Y and (1,2) Y are likewise known [162Jump To The Next Citation Point] 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 [57Jump To The Next Citation Point162Jump To The Next Citation Point] is only valid in a preferred scheme. For instance for the metric counter terms in this scheme additive contributions to Tij(𝔥) of the form ℒ 𝔥 V ij are absent [110Jump To The Next Citation Point]. Note that adding such a term for ν = 1 leaves the metric beta function in Equation (B.76View Equation) below unaffected, provided j V is functionally independent of 𝔥ij.

So far only the full fields entered, φjB on the bare and φj 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 j ξB (ξ) from the bare to the renormalized quantum fields was found by Howe, Papadopolous, and Stelle [110Jump To The Next Citation Point]. In effect one considers the inversion j ξ (¯φ;φ − φ¯) of the normal coordinate expansion φj = φj(¯φ;ξ) of the renormalized fields. If Z in Equation (B.61View Equation, B.66View Equation) is regarded as a differential operator acting on the second argument of this function, i.e. on φ,

ξj(ξ ) = Z ξj(¯φ;φ − ¯φ), (B.67 ) B
one obtains the desired ξjB(ξ) relation by inversion. To lowest order Z (1,1) = − 12∇2 yields
[ ] i i --1---λ- 1- i j 1- i k j 1-- i k j 3 ξB = ξ + 2 − d 2π 3R jξ + 4∇kR jξ ξ − 24∇ Rkjξ ξ + O(ξ ) . (B.68 )
At each loop order the coefficient is a power series in ξ whose coefficients are covariant expressions built from the metric 𝔥ij(¯φ) at the background point.

With all these renormalizations performed the result can be summarized in the proposition [110162Jump To The Next Citation Point] that the source-extended background functional

∫ { 1 ∫ } exp − Γ [φ¯; 𝒢] = [𝒟 ξ] exp − S [𝒢B, φB ] +- ddx Ji(φ¯)ξi (B.69 ) λ
defines a finite perturbative measure to all orders of the loop expansion. The additional source J (¯φ) i here is constrained by the requirement that ⟨ξj⟩ = 0. The key properties of Γ (φ¯; 𝒢) are:

Let V (φ), Vi(φ ), Vij(φ) 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 [162Jump To The Next Citation Point]

[[V (φ)]] = λV ⋅ -∂-LB = μd−2Z (𝔥)V, ∂F ∂ [ ∂ ] [[Vi(φ )∂ μφi]] = λVi ⋅----LB = μd− 2 ∂μφiZV (𝔥 )jiVj + Vi ⋅----Y , (B.70 ) ∂Vμi ∂V μi [[ 1 ]] ∂ μd− 2 [ √ -- √ -- ∂ ] -Vij(φ )∂μφi ∂μφj = λV ⋅ --LB − -√---∂μ ¯γ∂μφiN jik(𝔥)Vjk + ¯γVij ⋅--------Y . 2 ∂𝔥 ¯γ ∂(∂μ𝔥ij)
The functional derivatives here act on functionals on the target manifold at fixed x, e.g. -∂- ∫ D √ -- --∂--- V ⋅∂F = d φ 𝔥V (φ;x )∂F(φ;x). For 𝔥ij in addition the dependence of the counter terms on -- ∂ μ𝔥ij has to be taken into account, so that -- V ⋅-∂ := Vij ⋅-∂-+ ∂ μVij ⋅---∂--- ∂𝔥 ∂𝔥ij ∂(∂μ𝔥ij). Further LB = L (𝒢B, φB ) 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 ¯γμν. The additional total divergence in the last relation of Equation (B.70View Equation) reflects the effect of operator mixing. The normal products as given in Equation (B.70View Equation) 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 x-independent again to get the composite operators e.g. for the purely metric sigma-model.

The definition (B.70View Equation) 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

(ZV )j∂ V = ∂ (ZV ), (∂μZ )V = ∂ V ⋅-∂Y-- (B.71 ) i j i i ∂V μi
for a scalar V (φ;x). They entail
i -- ∂μ[[V ]] = [[∂iV∂ μφ ]] + [[∂ μV]]. (B.72 )
Moreover the invariance of the regularization under reparameterizations of the target manifold allows one to convert the reparameterization invariance of the basic Lagrangian (B.57View Equation) into a “diffeomorphism Ward identity” [201Jump To The Next Citation Point162Jump To The Next Citation Point]:
1 √ -- [[ 1 1 ]] δS √---∂μ[[ ¯γλJ μ(v)]] = -ℒv𝔥ij ∂ μφi∂μφj + ∂ μφiℒvVμi +-R (2)(¯γ)ℒv Φ + ℒvF − λvi ⋅--Bi ,(B.73 ) ¯γ 2 2 δφ
with λJ μ(v) = ∂μφivi + viV μi. The Lie derivative terms on the right-hand-side are the response of the couplings/sources under an infinitesimal diffeomorphism j j j φ → φ + v (φ ). Thus Jμ(v) 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.73View Equation) the following useful consistency conditions arise
[ ] 𝔥Bijvj = μd− 2 ZV (𝔥)jivj + Nijk(𝔥)ℒv𝔥jk , [ ] (B.74 ) viV B = μd− 2 ℒv 𝔥ij ⋅--∂Yμ----+ vi ⋅ ∂Y--+ Z (viV μi) . μi ∂ (∂ 𝔥ij) ∂V μi

So far the renormalization was done at a fixed normalization scale μ. The scale dependence of the renormalized couplings/sources 𝒢 = {𝔥ij,Vμi,D, F} is governed by a set of renormalization functions which follow from Equation (B.61View Equation). For a counter tensor of the form (B.62View Equation) it is convenient to introduce

∑ ( λ )l 𝒪˙ = − --- l𝒪 (1,l), (B.75 ) l≥1 2π
which in view of Equation (B.62View Equation) can be regarded as a parametric derivative of (1) 𝒪. Then
d μ ---𝔥ij = βij := (2 − d)𝔥ij − T˙ij, dμ d-- V : ˙V j ˙jk-- μ dμV μi = γ = (2 − d)Vμi − (Z )iVμj − Ni ∂μ 𝔥jk, (B.76 ) μ d-D = γD := (2 − d − ˙Z)D − U˙, dμ d μ --F = γF := (2 − d − ˙Z)F − ˙Y. dμ

The associated renormalization group operator is

𝒟 = μ ∂--+ β ⋅ ∂--+ γV ⋅ -∂--+ γD ⋅-∂--+ γF ⋅ -∂-. (B.77 ) ∂μ ∂𝔥 μ ∂Vμ ∂D ∂F
For example the dimension 0 composite operators in Equation (B.70View Equation) obey
𝒟 [[V (φ )]] = [[(d − 2 + Z˙+ 𝒟)V ]], (B.78 )
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 [201Jump To The Next Citation Point220Jump To The Next Citation Point162Jump To The Next Citation Point]

μν 1 μν i j 1 (2) D ¯γ [[T μν]] = 2[[Bij(𝔥∕λ)¯γ ∂μφ ∂νφ ]] + 2R (¯γ)[[B ]]. (B.79 )
Here the so-called Weyl anomaly coefficients enter:
|| λBij (𝔥∕λ) := λβij(𝔥∕λ) d=2 + ℒS 𝔥ij, D D | j (B.80 ) λB (D, 𝔥∕λ) := λγ (𝔥∕λ )|d=2 + S ∂jD,
where βij and D γ are the renormalization group functions of Equation (B.76View Equation) and
( λ )3 1 Si := Wi + ∂iD with Wi := N (1)(𝔥 )ijk𝔥jk = --- --∂i(RklmnRklmn ) + O(λ4). (B.81 ) 2π 32
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 δSB δφj have been omitted. The normal-products (B.70View Equation) are normalized such that the expectation value of an operator contains as its leading term the value of the corresponding functional on the background, ⟨𝒪 (φ)⟩ = 𝒪 (¯φ) + ..., where the subleading terms are in general nonlocal and depend on the scale μ. For the expectation value of the trace anomaly this produces a rather cumbersome expression (see e.g. [220]). As stressed in [201Jump To The Next Citation Point] the result (B.70View Equation), in contrast, allows one to use Bij(𝔥) = 0 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

φj −→ φj + --1--V j(φ, λ), (B.82 ) B B 2 − d
with V j(φ,λ ) = ∑ ( λ-)lV j(φ) l≥1 2π l functionally independent of the metric. Roughly speaking Equation (B.82View Equation) 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 Wj → Wj − Vj [201Jump To The Next Citation Point]. It is important to distinguish these diffeomorphisms from field renormalizations like Equation (3.65View Equation, 3.70View Equation) that depend on the metric. Although formally Equation (B.82View Equation) amounts to Ξj (φ,λ) −→ Ξj(φ,λ) + V j(φ,λ) in Equation (3.65View Equation); clearly one cannot cancel one against the other. The distinction is also highlighted by considering the change in the metric counter terms
(1) (1) Tij (𝔥) −→ Tij (𝔥) − ℒV 𝔥ij, (B.83 )
under Equation (B.82View Equation). Without further specifications this would not be legitimate for a 𝔥-dependent vector. Although the Lie derivative term in Equation (B.83View Equation) drops out when recomputing βij directly as a parametric derivative, in combinations like
μ i j μ i j --1--- μ i j βij(φB) ∂ φB ∂μφ B = βij(φ )∂ φ ∂μφ + 2 − d ℒV βij(φ) ∂ φ ∂ μφ + ... (B.84 )
the term (2 − d)𝔥ij in the metric beta function of Equation (B.76View Equation) induces an effective shift
βij(𝔥) −→ βij(𝔥) + ℒV 𝔥ij. (B.85 )
Similarly Wi is shifted to Wi − Vi and the Weyl anomaly coefficients are invariant.

In the context of Riemannian sigma-models D is usually interpreted as a “string dilaton” for the systems (B.57View Equation) defined on a curved base space. If one is interested in the renormalization of Equation (B.57View Equation) on a flat base space, D on the other hand plays the role of a potential for the improvement term of the energy momentum tensor. This role of D can be made manifest by rewriting Equation (B.79View Equation) by means of the diffeomorphism Ward identity. Returning to a flat base space one finds [201162]

[[T μ ]] = 1[[β (𝔥∕λ) ∂μφi∂ φj]] + ∂μ∂ [[D ]] + ∂μ[[∂ φiW ]], (B.86 ) μ 2 ij μ μ μ i
where again terms proportional to the equations of motion operator have been omitted. Here ∂2[[D ]] is the ‘naive’ improvement term while the additional total divergence is induced by operator mixing.

The functions BD and Bij(g) are linked by an important consistency condition, the Curci–Paffuti relation [57]. We present it in two alternative versions,

∂ ∂i ˙U = N˙ijk T˙jk − ˙T ⋅--Wi + (Z˙V )jiWj, ∂g (B.87 ) D ˙ jk ∂-- j ∂iB = Ni Bjk − B ⋅ ∂g Si + BijS .
The first version displays the fact that the identity relates various 𝔥-dependent counter terms without D entering. In the second version D is introduced in a way that yields an identity among the Weyl anomaly coefficients. It has the well-known consequence that BD is constant when Bij vanishes:
D Bij = 0 =⇒ B = cT∕6, (B.88 )
where cT is the central charge of energy momentum tensor derived from Equation (B.57View Equation).

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